# Schrödinger Equation Driven by the Square of a Gaussian Field: Instanton
Analysis in the Large Amplification Limit

Philippe Mounaix<EMAIL_ADDRESS>CPHT, CNRS, École
polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France.

###### Abstract

We study the tail of $p(U)$, the probability distribution of
$U=|\psi(0,L)|^{2}$, for $\ln U\gg 1$, $\psi(x,z)$ being the solution to
$\partial_{z}\psi-\frac{i}{2m}\nabla_{\perp}^{2}\psi=g|S|^{2}\,\psi$, where
$S(x,z)$ is a complex Gaussian random field, $z$ and $x$ respectively are the
axial and transverse coordinates, with $0\leq z\leq L$, and both $m\neq 0$ and
$g>0$ are real parameters. We perform the first instanton analysis of the
corresponding Martin-Siggia-Rose action, from which it is found that the
realizations of $S$ concentrate onto long filamentary instantons, as $\ln
U\to+\infty$. The tail of $p(U)$ is deduced from the statistics of the
instantons. The value of $g$ above which $\langle U\rangle$ diverges coincides
with the one obtained by the completely different approach developed in
Mounaix et al. 2006 Commun. Math. Phys. 264 741\. Numerical simulations
clearly show a statistical bias of $S$ towards the instanton for the largest
sampled values of $\ln U$. The high maxima — or ‘hot spots’ — of
$|S(x,z)|^{2}$ for the biased realizations of $S$ tend to cluster in the
instanton region.

stochastic partial differential equations, instanton analysis, extreme event
statistics, laser-plasma interactions

##  I Introduction

In the second part of their seminal paper on the breakdown of linear
instability in stimulated Brillouin scattering RD1994 , Rose and DuBois
investigated the following equation for the complex amplitude $\psi(x,z)$ of
the scattered light electric field

$\left\\{\begin{array}[]{l}\partial_{z}\psi(x,z)-\frac{i}{2m}\nabla_{\perp}^{2}\psi(x,z)=g|S(x,z)|^{2}\psi(x,z),\\\
0\leq z\leq L,\ x\in\Lambda\subset\mathbb{R}^{d},\ {\rm and}\
\psi(x,0)=1.\end{array}\right.$ (1)

In Eq. (1), $z$ and $x$ respectively denote the axial and transverse
coordinates in a plasma of length $L$ and cross-sectional domain $\Lambda$
(often a torus like, e.g., in mathematics oriented work and/or numerical
simulations using spectral methods). The boundary condition at $z=0$ is taken
to be a constant for simplicity and $m\neq 0$ is a real parameter introduced
for convenience. In Ref. RD1994 , the coupling constant $g>0$ is proportional
to the average laser intensity and the complex amplitude of the laser electric
field $S(x,z)$ is a homogeneous Gaussian random field with zero mean and
normalized intensity $\langle|S(x,z)|^{2}\rangle=1$. For our purposes, we can
be less restrictive and take $S(x,z)$ transversally homogeneous with
normalization $L^{-1}\int_{0}^{L}\langle|S(x,z)|^{2}\rangle\,dz=1$. From now
on, we accept the idealizations inherent in the derivation of Eq. (1), setting
aside the question of its validity as a realistic model, which varies from one
physical problem to the other. As a stochastic PDE, the diffraction-
amplification problem (1) is a Schrödinger equation driven by the square of a
Gaussian field.

Using heuristic arguments and numerical simulations, Rose and DuBois found
that the expected value of the scattered energy density,
$\langle|\psi(x_{0},L)|^{2}\rangle$, at some given $x_{0}\in\Lambda$ diverges
for every $L>0$ when $g$ is greater than some critical value, $g_{c}(L)$, yet
to be determined. Here, the average $\langle|\psi|^{2}\rangle$ is taken over
the realizations of the Gaussian field $S$. Physically, this divergence was
interpreted in RD1994  as indicating the breakdown of the linear model (1) and
the onset of a saturated nonlinear regime in high overintensities, or hot
spots, of $|S(x,z)|^{2}$. We will shortly come back to the role of the hot
spots in the divergence of $\langle|\psi(x_{0},L)|^{2}\rangle$. Note that in
the limit referred to in RD1994  as the independent hot spot model, this
divergence was pointed out by Akhmanov et al. 20 years before ADP1974 . The
problem was then analyzed in ADLM2001 ; ML2004 ; MCL2006  from a more rigorous
mathematical point of view, establishing the numerical results of Ref. RD1994
on much firmer ground and giving the exact expression of the critical coupling
$g_{c}(L)$. In the following, we will take $x_{0}=0$ without loss of
generality (by statistical invariance under $x$-translation) and we will write
$U=|\psi(0,L)|^{2}$.

Whether or not $\langle U\rangle$ diverges depends on the extreme upper tail
of $p(U)$ — the probability distribution function (PDF) of $U$ — in the limit
of a large $\ln U$ (see the beginning of Sec. II). It is then natural to ask
what the realizations of $S(x,z)$ yielding a large $\ln U$ are like, with $U$
and $S(x,z)$ related to each other through Eq. (1), from which probability
distribution they are drawn, and if the corresponding tail of $p(U)$ does give
the correct value of $g_{c}(L)$. Answering these questions is the subject of
this paper.

To put our work into perspective, it is interesting to recall how the
existence of $g_{c}(L)$ has been interpreted in laser-plasma physics
literature since Ref. RD1994 . The interpretation relies on the implicit
assumption that the realizations of $|S(x,z)|^{2}$ giving rise to a large $\ln
U\gg\langle\ln U\rangle$ and the generic ones for which $\ln U\simeq\langle\ln
U\rangle$ are alike, in the sense of being made up of local, statistically
independent, overintensities, or hot spots, separated from each other by a few
correlation lengths of $S(x,z)$ Dixit1993 ; G1985 ; RD1993 . Hot spot
contribution to the amplification of $|\psi|^{2}$ can then be computed by
using the remarkable result that intense hot spots have a non-random profile
depending on the correlation function of $S(x,z)$ and being the same for each
hot spot RD1993 ; A1981 . Thus, intense hot spots are entirely characterized
by their random intensity which turns out to be exponentially distributed (for
large intensity and to within slow, algebraic, corrections) RD1993 ; G1999 .
For $g$ large enough, intense hot spots become statistically significant as
the exponentially large amplification they produce outbalances their
exponentially small scarcity, leading to the divergence of $\langle U\rangle$.
The smallest value of $g$ at which this divergence occurs defines the critical
coupling $g_{c}(L)$ and for $g>g_{c}(L)$ physics could be expected to be
dominated by intense hot spots. Unfortunately, this interpretation fails to
give the correct value of $g_{c}(L)$ for $L$ greater than a hot spot length
Mounaix2001 ; MD2004 . The assumption of high intensity, statistically
independent hot spots giving the dominant contribution to the amplification in
the large $\ln U$ limit must be revisited. Large values of $\ln U$ are
produced by rare realizations of $S(x,z)$ that have no reason a priori to look
like generic realizations with no other structures than uncorrelated, local
hot spots randomly scattered in $\Lambda\times[0,L]$. It may or may not be so:
the answer will come out of the calculations.

In the simpler diffraction-free case where $m^{-1}=0$ in Eq. (1), the problem
reduces to a mere $1D$ amplification along $z$ with
$\psi(L)=\exp\left(g\int_{0}^{L}|S(z)|^{2}dz\right)$. A large value of
$\ln|\psi(L)|^{2}$ corresponds to a large value of $\int_{0}^{L}|S(z)|^{2}dz$.
Thus, the realizations of $S(z)$ that form the tail of $p(U)$ are the ones
with a large $L^{2}$-norm. These realizations were studied thoroughly in
MD2004 ; MC2011 ; MMB2012 . Let $C(z,z^{\prime})=\langle
S(z)S(z^{\prime})^{\ast}\rangle$ and define the covariance operator $T_{C}$
acting on $f(z)\in L^{2}([0,L])$ by

$(T_{C}f)(z)=\int_{0}^{L}C(z,z^{\prime})\,f(z^{\prime})\,dz^{\prime},$ (2)

with $0\leq z\leq L$. As a correlation function, $C$ is a positive definite
kernel and all the eigenvalues of $T_{C}$ are necessarily real and positive.
Write $\mu_{1}>0$ the largest eigenvalue of $T_{C}$ with degeneracy $d_{1}$.
It was proved in MD2004 ; MC2011  that the realizations of $S(z)$ with a large
$L^{2}$-norm concentrate onto the fundamental eigenspace of $T_{C}$, i.e., the
eigenspace associated with the largest eigenvalue $\mu_{1}$. More
specifically, writing $\\{\phi_{1},\cdots,\phi_{d_{1}}\\}$ an orthonormal
basis of the fundamental eigenspace of $T_{C}$, one has

$S(z)\sim\sqrt{\eta}\sum_{i=\nu}^{d_{1}}a_{\nu}\phi_{\nu}(z)\ \ \ \ \
(\|S\|_{2}\to+\infty),$ (3)

with $\eta\sim\|S\|_{2}^{2}$, where $\|\cdot\|_{2}$ denotes the $L^{2}$-norm
over $[0,L]$. The $a_{\nu}$s are complex numbers normalized to
$\sum_{\nu=1}^{d_{1}}|a_{\nu}|^{2}=1$. The probability distribution of $\eta$
has the gamma-distribution tail $p(\eta)\sim\eta^{d_{1}-1}{\rm
e}^{-\eta/\mu_{1}}$ for large $\eta$, and the $a_{\nu}$s define a random
$2d_{1}$-dimensional (real) unit vector $\bm{a}$ with coordinates ${\rm
Re}(a_{\nu})$ and ${\rm Im}(a_{\nu})$ ($1\leq\nu\leq d_{1}$) the direction of
which is uniformly distributed over the unit $(2d_{1}-1)$-sphere. From Eq. (3)
it is clear that the realizations of $S(z)$ with a large $L^{2}$-norm are less
random than the Gaussian field $S(z)$ itself. It only takes $2d_{1}$ random
quantities to characterize these realizations entirely: $\eta$ and the
direction of $\bm{a}$. For instance, if $\mu_{1}$ is not degenerate
($d_{1}=1$), Eq. (3) yields

$\frac{S(z)}{\|S\|_{2}}\sim{\rm e}^{i\theta}\phi_{1}(z)\ \ \ \ \
(\|S\|_{2}\to+\infty),$ (4)

where $\theta$ is a random phase uniformly distributed over $[0,2\pi)$ and
$|S(z)|/\|S\|_{2}\sim|\phi_{1}(z)|$ is non-random, which means that the
profile of $S(z)$ is purely deterministic in this case. Note that Eq. (4)
rules out any description in terms of localized hot spots when $L$ is large,
as $\phi_{1}(z)$ typically is a one-bump delocalized mode spreading over the
whole domain $0\leq z\leq L$ (see MD2004  for details). As will be seen
further on, the randomness reduction of $S$ when the amplification is large
occurs in the $m^{-1}\neq 0$ case too.

From $U=|\psi(L)|^{2}=\exp(2g\|S\|_{2}^{2})$ and $\eta\sim\|S\|_{2}^{2}$ as
$\|S\|_{2}\to+\infty$, one gets $\eta\sim(2g)^{-1}\ln U$ as $\ln U\to+\infty$.
The tail of $p(U)$ is then readily obtained from
$p(\eta)\sim\eta^{d_{1}-1}{\rm e}^{-\eta/\mu_{1}}$ and the change of variables
from $\eta$ to $U$. One finds, in logarithmic form,

$\displaystyle\ln p(U)$ $\displaystyle\sim$
$\displaystyle\ln\left[\frac{1}{U}\,p\left(\eta=\frac{1}{2g}\ln
U\right)\right]$ $\displaystyle=$
$\displaystyle-\left(1+\frac{1}{2\mu_{1}g}\right)\,\ln
U+\left(d_{1}-1\right)\,\ln\ln U+O(1)\ \ \ \ \ (\ln U\to+\infty),$

from which it follows that $p(U)$ has a leading algebraic tail $\propto
U^{-\zeta}$ (modulated by logarithmic corrections in the amplitude) with
exponent $\zeta=(1+1/2\mu_{1}g)$ depending continuously on the parameters of
the model. Injecting this result into $\langle
U\rangle=\int_{1}^{+\infty}Up(U)\,dU$, one finds that the critical coupling in
the diffraction-free case is given by $g_{c}(L)=1/2\mu_{1}$ (where $\mu_{1}$
depends on $L$) MD2004 . Note that it is also possible to determine the tail
of $p(U)$ exactly from the full Gaussian statistics of $S$, which makes it
possible to estimate the contribution of the subleading corrections to Eq.
(3). Skipping the details, one finds that these corrections do not contribute
to $\ln p(U)$ by terms greater than $O(1)$ as $\ln U\to+\infty$.

To conclude this brief overview of diffraction-free results, let us mention
the interesting connection between the concentration onto the fundamental
eigenspace of $T_{C}$ in Eq. (3) and the Bose-Einstein condensation of $S(z)$
in the ‘thermodynamic’ limit defined by $L\to+\infty$ and
$\|S\|_{2}^{2}\to+\infty$ with fixed $\|S\|_{2}^{2}/L$, see MMB2012 . Note
also that more general concentration properties can be found in the limit
where the large $L^{2}$-norm is replaced with a large quadratic or linear
form, see Mounaix2015 ; Mounaix2019 .

By contrast, much less is known in the general case with diffraction where
$m^{-1}\neq 0$ in Eq. (1). The only results so far are the numerical ones in
the second part of RD1994  and the analytical calculation of the critical
coupling performed in MCL2006 , where it is proved that the critical coupling
without diffraction cannot be less than the one with diffraction, the latter
being given by

$g_{c}(L)=\frac{1}{2\sup_{x(\cdot)\in B(0,L)}\mu_{1}[x(\cdot)]},$ (6)

and the former by $1/2\mu_{1}[x(\cdot)\equiv 0]$. In Eq. (6), $B(0,L)$ denotes
the set of all the continuous paths in $\Lambda$ satisfying $x(L)=0$ and
$\mu_{1}[x(\cdot)]$ is the largest eigenvalue of the covariance operator
$T_{x(\cdot)}$ defined by Eq. (2) with $C(z,z^{\prime})=\langle
S(x(z),z)S(x(z^{\prime}),z^{\prime})^{\ast}\rangle$.

The question then arises whether the non-local quantity $\sup_{x(\cdot)\in
B(0,L)}\mu_{1}[x(\cdot)]$ in the expression for $g_{c}(L)$ is the signature of
a corresponding non-local structure in the realizations of $S(x,z)$ giving
rise to a large $\ln U$. (Note that only local quantities computed on the hot
spot scale can appear in the hot spot model.) To answer this question we need
to find a way to identify such realizations. The corresponding tail of $p(U)$
will then be tested in return by checking that the critical coupling it yields
coincides with the one in Eq. (6). The calculations in MD2004 ; MC2011  are of
no help as being specific to the diffraction-free case. To determine $S(x,z)$
when $\ln U$ is large and get the tail of $p(U)$ in the presence of
diffraction we need a different approach.

A possible line of attack is through the functional integral formalism
introduced by Janssen Janssen1976 , DeDominicis DeDominicis1976 ; DDP1978 ,
and Phythian Phythian1977  (see also JP1979 ; Jensen1981 ). This method
provides a formal description of classical statistical dynamics in terms of
functional integrals analogous to Feynman’s action-integral formalism of
quantum theory. Applying the method to the stochastic equation (1), one finds
that $p(U)$ can be formally written as the functional integral

$p(U)=\int_{\varphi(x,0)=1}\delta(U-|\varphi(0,L)|^{2})\,{\rm
e}^{\mathcal{A}}\,\mathscr{D}^{2}\varphi\,\mathscr{D}^{2}\tilde{\varphi}\,\mathscr{D}^{2}S,$
(7)

where $\varphi$ and $\tilde{\varphi}$ are complex Martin-Siggia-Rose conjugate
fields MSR1973 , $\mathscr{D}^{2}\equiv\mathscr{D}{\rm
Re}(\cdot)\mathscr{D}{\rm Im}(\cdot)$, and
$\mathcal{A}\equiv\mathcal{A}(\varphi,\tilde{\varphi},S)$ is an ‘action’
depending on $\varphi$, $\tilde{\varphi}$, and $S$, often referred to as the
Martin-Siggia-Rose (MSR) action in the literature. When $\ln U$ is large, the
functional integral on the right-hand side of Eq. (7) is determined by the
field configuration — called the ‘leading instanton’ — corresponding to the
highest saddle-point of the action $\mathcal{A}$. Note that, usually, $S$ is
integrated out in Eq. (7), leading to

$p(U)=\int_{\varphi(x,0)=1}\delta(U-|\varphi(0,L)|^{2})\,{\rm
e}^{\mathcal{L}}\,\mathscr{D}^{2}\varphi\,\mathscr{D}^{2}\tilde{\varphi},$ (8)

where the action $\mathcal{L}\equiv\mathcal{L}(\varphi,\tilde{\varphi})$ is
defined by

${\rm e}^{\mathcal{L}(\varphi,\tilde{\varphi})}=\int{\rm
e}^{\mathcal{A}(\varphi,\tilde{\varphi},S)}\,\mathscr{D}^{2}S.$ (9)

The expressions of $p(U)$ in Eqs. (7) and (8) are equivalent. Either can be
used and the large $\ln U$ behavior of $p(U)$ can equally be obtained from an
instanton analysis of Eq. (8), instead of Eq. (7). As we want to determine the
most probable realizations of $S(x,z)$ when $\ln U$ is large, it is natural to
keep $S$ explicit and to use the functional representation (7).

At large $\ln U$ and fixed instanton, the fluctuations of the fields around
the instanton are small and can be integrated out in Eq. (7) as standard
Gaussian fluctuations, yielding the leading asymptotic behavior

$\ln p(U)\sim\ln\left\langle\delta(U-|\varphi_{\rm
inst}(0,L)|^{2})\right\rangle_{S_{\rm inst}}\ \ \ \ \ (\ln U\to+\infty),$ (10)

where the subscript ‘inst’ stands for leading instanton and
$\langle\cdot\rangle_{S_{\rm inst}}$ denotes the average over the realizations
of $S_{\rm inst}$. In the diffraction-free case, it will be checked in Sec. II
that $S_{\rm inst}$ coincides with the right-hand side of Eq. (3) and that the
tail of $p(U)$ in Eq. (10) is the same as the one in Eq. (I).

One of the early attempts to obtain PDF tails from the highest saddle-point of
the action in a functional integral representation was made by Giles for
Navier-Stokes turbulence Giles1995 . Unfortunately, the perturbative approach
followed in this work was doomed to fail as instantons are nonperturbative
objects by nature. The nonperturbative instanton analysis of intermittency in
fluid turbulence was initiated by Falkovich et al. in FKLM1996 , but many
questions are still open SGMG2022 ; AMPV2022 . Since then, advances and new
applications of the instanton approach in stochastic field theories have been
made. A relatively recent review on fluid turbulence applications can be found
in GGS2015 . Different kinds of nonlinear Schrödinger equations with additive
noise have been analyzed using instanton calculus within the last twenty
years, see e.g. FKLT2001 , among others. See also the instanton analyses of
PDF tails in forced Burgers turbulence in GM1996 ; BM1996  and the numerical
results in GGS2013 . Instanton solutions have been tested numerically for the
Kardar-Parisi-Zhang equation in HMS2019 . Here, we give the first instanton
analysis of the stochastic amplifier (1) in the large amplification limit of
interest in the overcritical regime $g>g_{c}(L)$. Our strategy is two-step:

  * $\bullet$

write the MSR action $\mathcal{A}(\varphi,\tilde{\varphi},S)$ for the
stochastic amplifier (1) and find the corresponding leading instanton
$S_{\rm{inst}}$. The realizations of $S_{\rm{inst}}$ define the driver onto
which the realizations of $S$ concentrate in the large $\ln U$ limit, which
generalizes the diffraction-free result in Eq. (3) to the case with
diffraction;

  * $\bullet$

use the instanton $\varphi_{\rm{inst}}$ and $S_{\rm{inst}}$ obtained at the
first step on the right-hand side of Eq. (10) to get the tail of $p(U)$.

Before entering the details of the calculations, it is useful to give a brief
summary of the main new results obtained in this paper.

  * $\diamond$

We show that in the large $\ln U$ limit, the realizations of $S$ concentrate
onto large-scale filamentary instantons running along specific non-random
paths in $B(0,L)$. Each of these paths, denoted by $x_{\rm inst}(\cdot)$, is a
path maximizing the largest eigenvalue $\mu_{1}[x(\cdot)]$ of the covariance
operator $T_{x(\cdot)}$ defined in Eq. (43). In the case of a ‘single-filament
instanton’ (see Sec. IV) and assuming a non-degenerate $\mu_{1}[x_{\rm
inst}(\cdot)]$, we prove that

$S(x,z)\sim\frac{c_{1}}{\mu_{\rm max}}\,\int_{0}^{L}C(x-x_{\rm
inst}(z^{\prime}),z,z^{\prime})\,\phi_{1}(z^{\prime})\,dz^{\prime}\ \ \ \ \
(\ln U\to+\infty),$ (11)

  * where $C(x-x^{\prime},z,z^{\prime})=\langle S(x,z)S(x^{\prime},z^{\prime})^{\ast}\rangle$, $\mu_{\rm max}=\mu_{1}[x_{\rm inst}(\cdot)]$ with normalized eigenfunction $\phi_{1}$, and $c_{1}$ is a complex Gaussian random variable with $\langle c_{1}\rangle=\langle c_{1}^{2}\rangle=0$ and $\langle|c_{1}|^{2}\rangle=\mu_{\rm max}$. The instanton on the right-hand side of Eq. (11) lives within a long thin tube, or filament, running along $x_{\rm inst}(\cdot)$ (see the end of Sec. IV.1). Normalizing $S$ to its $L^{2}$-norm, Eq. (11) yields

$\frac{S(x,z)}{\|S\|_{2}}\sim A{\rm e}^{i\arg(c_{1})}\int_{0}^{L}C(x-x_{\rm
inst}(z^{\prime}),z,z^{\prime})\,\phi_{1}(z^{\prime})\,dz^{\prime}\ \ \ \ \
(\ln U\to+\infty),$

  * where $A>0$ is a constant, and the profile of $S(x,z)$ defined by $|S(x,z)|/\|S\|_{2}$ is asymptotically non-random as $\ln U\to+\infty$.

  * $\diamond$

We determine the tail of $p(U)$ for large $\ln U$ from the statistics of the
instanton on the right-hand side of Eq. (11). We find that $p(U)$ has a
leading algebraic tail $\propto U^{-\zeta}$ with exponent
$\zeta=(1+1/2\mu_{\rm max}g)$, modulated by a slow varying amplitude (slower
than algebraic). Injecting this result into $\langle
U\rangle=\int_{1}^{+\infty}Up(U)\,dU$, we find that $\langle U\rangle$
diverges for all $g>1/2\mu_{\rm max}$. The critical coupling is thus given by
$g_{c}(L)=1/2\mu_{\rm max}$, where $\mu_{\rm max}$ depends on $L$, in
agreement with Eq. (6). We can then explain the intriguing presence of the
non-local quantity $\mu_{\rm max}$ in the expression of $g_{c}(L)$ as a direct
consequence of the fact that the realizations of $S$ causing the divergence of
$\langle U\rangle$ are realizations of the non-local instanton (11), rather
than of localized hot spots, as is widely assumed.

  * $\diamond$

Finally, the emergence of the instanton in the realizations of $S$ as $\ln U$
increases is observed in numerical simulations as a statistical bias of $S$
towards the instanton. For the largest sampled values of $\ln U$, the emerging
large-scale instanton coexists with small-scale hot spots in a non-negligible
fraction of realizations. The presence of the instanton causes the hot spots
to cluster in the instanton region instead of being uniformly scattered in
$\Lambda\times[0,L]$, and the level of $|S(x,z)|^{2}$ between the hot spots
remains significantly higher than it would be in the absence of instanton.

The paper is organized as follows. In Section II, we test the functional
approach by revisiting the diffraction-free problem where the results are
already known. In Section III, we write the instanton equations for the full
problem with diffraction in the case of one transverse dimension ($d=1$) and
we specify the class of $S$ we consider. Section IV is devoted to the solution
of the instanton equations in the case of ‘single-filament’ instantons. The
corresponding tail of $p(U)$ is determined. In Section V, we report on
numerical simulations for realizations of $S$ in a sample of experimentally
realistic size. Finally, we discuss our results and their implications,
especially in laser-matter interaction physics, and we give potential
perspectives in Section VI. Some technical material is relegated to the
appendices.

##  II Amplification without diffraction revisited

As a warm-up to the full problem (1), we test the functional approach on the
simpler problem without diffraction and see how the results in Eqs. (3) and
(I) can also be obtained from an instanton analysis of the appropriate MSR
action.

Before we start, it is useful to briefly come back to the definition of the
asymptotic limit. In the absence of diffraction,
$U=\exp\left(2g\|S\|_{2}^{2}\right)$ and the limit $\|S\|_{2}^{2}\to+\infty$
in MD2004 ; MC2011  reads $\ln U\to+\infty$, which defines the asymptotic
limit (rather than $U\to+\infty$). The same applies to the case with
diffraction, where $U$ is also the result of exponential amplification.

###  II.1 The MSR action $\bm{\mathcal{A}(\varphi,\tilde{\varphi},S)}$

In the diffraction-free limit, $m^{-1}=0$, the equation (1) reduces to the
$1D$ stochastic amplifier (for fixed $x$, not written)

$\left\\{\begin{array}[]{l}d_{z}\psi(z)-g|S(z)|^{2}\psi(z)=0,\\\ 0\leq z\leq
L\ {\rm and}\ \psi(0)=1.\end{array}\right.$ (12)

Let $F[\psi(\cdot)]$ be a functional of $\psi(z)$ solution to Eq. (12). From
the general formalism developed in Janssen1976 ; DeDominicis1976 ; DDP1978 ;
Phythian1977 ; JP1979 ; Jensen1981  it can be shown that $F[\psi(\cdot)]$
admits the functional integral representation

$F[\psi(\cdot)]=\int_{\varphi(0)=1}F[\varphi(\cdot)]\,{\rm
e}^{\frac{i}{2}\,\left(\left\langle\tilde{\varphi}\left|d_{z}-g|S|^{2}\right|\varphi\right\rangle+c.\,c.\right)}\,\mathscr{D}^{2}\varphi\,\mathscr{D}^{2}\tilde{\varphi},$
(13)

with Dirac’s bracket notation $\langle
f|O|h\rangle=\int_{0}^{L}f(z)^{\ast}(Oh)(z)\,dz$. Note that since $\psi(0)$ is
real, $\psi(z)$ is also real for all $z$ and a representation with real
$\varphi$ and $\tilde{\varphi}$ would have been sufficient. In Eq. (13) we
have kept complex $\varphi$ and $\tilde{\varphi}$ in anticipation of the
generalization to the case with diffraction. Now, using (13) with
$F[\psi(\cdot)]=\delta(U-|\psi(L)|^{2})$ in
$p(U)=\left\langle\delta(U-|\psi(L)|^{2})\right\rangle_{S}$, where
$\langle\cdot\rangle_{S}$ denotes the average over the realizations of $S$,
one gets

$\displaystyle
p(U)=\int_{\varphi(0)=1}\left\langle\delta(U-|\varphi(L)|^{2})\,{\rm
e}^{\frac{i}{2}\,\left(\left\langle\tilde{\varphi}\left|d_{z}-g|S|^{2}\right|\varphi\right\rangle+c.\,c.\right)}\right\rangle_{S}\,\mathscr{D}^{2}\varphi\,\mathscr{D}^{2}\tilde{\varphi}$
$\displaystyle=\int_{\varphi(0)=1}\delta(U-|\varphi(L)|^{2})\,{\rm
e}^{\frac{i}{2}\,\left(\left\langle\tilde{\varphi}\left|d_{z}-g|S|^{2}\right|\varphi\right\rangle+c.\,c.\right)-\left\langle
S\left|T_{C}^{-1}\right|S\right\rangle}\,\mathscr{D}^{2}\varphi\,\mathscr{D}^{2}\tilde{\varphi}\,\mathscr{D}^{2}S,$
(14)

where $T_{C}$ is the covariance operator of $S$ defined in Eq. (2). The
functional integral representation of $p(U)$ in Eq. (II.1) is of the same form
as the one in Eq. (7) with MSR action

$\displaystyle\mathcal{A}(\varphi,\tilde{\varphi},S)=\frac{i}{2}\,\left(\left\langle\tilde{\varphi}\left|d_{z}-g|S|^{2}\right|\varphi\right\rangle+c.\,c.\right)-\left\langle
S\left|T_{C}^{-1}\right|S\right\rangle$
$\displaystyle=\frac{i}{2}\,\left[\int_{0}^{L}\tilde{\varphi}^{\ast}(z)\left(d_{z}\varphi(z)-g|S(z)|^{2}\varphi(z)\right)dz+c.\,c.\right]-\int_{0}^{L}S^{\ast}(z)(T_{C}^{-1}S)(z)\,dz.$
(15)

###  II.2 Leading instanton and tail of $\bm{p(U)}$

The leading instanton which determines the large $\ln U$ behavior of $p(U)$ in
Eq. (II.1) is a stationary point of $\mathcal{A}(\varphi,\tilde{\varphi},S)$
under the restriction $|\varphi(L)|^{2}=U$. According to the usual procedure
of Lagrange multipliers CH1989 , it can be found as a stationary point of the
action
$\mathcal{A}^{\prime}(\varphi,\tilde{\varphi},S)=\mathcal{A}(\varphi,\tilde{\varphi},S)+\lambda|\varphi(L)|^{2}$
without restriction, where $\lambda$ is a Lagrange multiplier. Write
$\delta\mathcal{A}^{\prime}(\varphi,\tilde{\varphi},S)$ the variation of
$\mathcal{A}^{\prime}(\varphi,\tilde{\varphi},S)$ under variations of the
fields and their complex conjugates treated as independent variables, with
endpoints $\varphi(0)=\varphi^{\ast}(0)=1$, and
$\tilde{\varphi}(L^{+})=\tilde{\varphi}^{\ast}(L^{+})=0$ (by causality
principle. See, e.g., Ref. FKLM1996 ). It is convenient to make the
independence of $\tilde{\varphi}$ and $\tilde{\varphi}^{\ast}$ explicit by
writing $\tilde{\varphi}^{\ast}=\tilde{\vartheta}$, independent of
$\tilde{\varphi}$. The stationarity condition
$\delta\mathcal{A}^{\prime}(\varphi,\tilde{\varphi},S)=0$ leads to the
equations

$\displaystyle d_{z}\varphi(z)-g|S(z)|^{2}\varphi(z)=0\ {\rm with}\
\varphi(0)=1,$ $\displaystyle
d_{z}\tilde{\varphi}(z)+g|S(z)|^{2}\tilde{\varphi}(z)=-2i\lambda\varphi(L)\delta(z-L)\
{\rm with}\ \tilde{\varphi}(L^{+})=0,$ (16) $\displaystyle
d_{z}\tilde{\vartheta}(z)+g|S(z)|^{2}\tilde{\vartheta}(z)=-2i\lambda\varphi^{\ast}(L)\delta(z-L)\
{\rm with}\ \tilde{\vartheta}(L^{+})=0,$

and

$(T_{C}^{-1}S)(z)=-\frac{ig}{2}\left[\tilde{\vartheta}(z)\varphi(z)+\tilde{\varphi}(z)\varphi^{\ast}(z)\right]S(z),$
(17)

or, equivalently,

$\displaystyle d_{z}\varphi(z)-g|S(z)|^{2}\varphi(z)=0\ {\rm with}\
\varphi(0)=1,$ $\displaystyle
d_{z}\tilde{\varphi}(z)+g|S(z)|^{2}\tilde{\varphi}(z)=0\ {\rm with}\
\tilde{\varphi}(L)=2i\lambda\varphi(L),$ (18) $\displaystyle
d_{z}\tilde{\vartheta}(z)+g|S(z)|^{2}\tilde{\vartheta}(z)=0\ {\rm with}\
\tilde{\vartheta}(L)=2i\lambda\varphi^{\ast}(L),$

and

$\left[T_{C}\left(\tilde{\vartheta}\varphi+\tilde{\varphi}\varphi^{\ast}\right)S\right](z)=\frac{2i}{g}S(z).$
(19)

The equations (II.2) are readily solved. One gets,

$\displaystyle\varphi(z)={\rm
e}^{g\int_{0}^{z}|S(z^{\prime})|^{2}dz^{\prime}},$
$\displaystyle\tilde{\varphi}(z)=2i\lambda\varphi(L){\rm
e}^{g\int_{z}^{L}|S(z^{\prime})|^{2}dz^{\prime}},$ (20)
$\displaystyle\tilde{\vartheta}(z)=2i\lambda\varphi^{\ast}(L){\rm
e}^{g\int_{z}^{L}|S(z^{\prime})|^{2}dz^{\prime}},$

and
$\tilde{\vartheta}(z)\varphi(z)=\tilde{\varphi}(z)\varphi^{\ast}(z)=4i\lambda|\varphi(L)|^{2}=4i\lambda
U$, independent of $z$. Injecting this solution onto the left-hand side of
(19), one obtains the eigenvalue equation

$(T_{C}S)(z)=\frac{1}{2\lambda gU}S(z).$ (21)

It follows immediately from Eq. (21) that an instanton solution for $S$ is an
eigenfunction of its covariance operator $T_{C}$, which fixes the value of
$\lambda$ for each instanton, namely $\lambda=1/(2\mu_{n}gU)$, where
$\mu_{1}>\mu_{2}>\cdots>0$ are the eigenvalues of $T_{C}$. Using (II.2) and
(21) in (II.1), one finds that the action of an instanton solution is
$\mathcal{A}=-\mu_{n}^{-1}\|S\|_{2}^{2}$, and the leading instanton $S_{\rm
inst}$ corresponds to the largest eigenvalue $\mu_{1}$ for which $\mathcal{A}$
is maximum. For simplicity, we assume that $\mu_{1}$ is not degenerate. The
generalization to a degenerate $\mu_{1}$ is straightforward and we leave it to
the reader as an exercice. Writing $\phi_{1}$ the normalized eigenfunction
associated with $\mu_{1}$, one gets

$S_{\rm inst}(z)=c_{1}\phi_{1}(z),$ (22)

where $c_{1}$ is a complex number. The other components of the leading
instanton, $\varphi_{\rm inst}$, $\tilde{\varphi}_{\rm inst}$, and
$\tilde{\vartheta}_{\rm inst}$, are the instanton solution in Eq. (II.2) with
$S=S_{\rm inst}$ and $\lambda=1/(2\mu_{1}gU)$. In the following, we will only
need the expression of $\varphi_{\rm inst}(L)$,

$\varphi_{\rm inst}(L)=\exp\left(g\|S_{\rm inst}\|_{2}^{2}\right),$ (23)

where $\|\cdot\|_{2}$ denotes the $L^{2}$-norm over $[0,L]$.

Integrating out the fluctuations around the instanton, at fixed instanton, in
Eq. (II.1) and using the expressions of $S_{\rm inst}$ and $\varphi_{\rm
inst}$ in Eqs. (22) and (23), one obtains the diffraction-free version of the
asymptotic expression (10),

$\displaystyle\ln p(U)$ $\displaystyle\sim$
$\displaystyle\ln\int\delta\left(U-{\rm e}^{2g|c_{1}|^{2}}\right)\,{\rm
e}^{-|c_{1}|^{2}/\mu_{1}}\,\frac{d^{2}c_{1}}{\pi\mu_{1}}$ (24)
$\displaystyle=$ $\displaystyle\ln\int_{0}^{+\infty}\delta\left(U-{\rm
e}^{2g\eta}\right)\,{\rm e}^{-\eta/\mu_{1}}\frac{d\eta}{\mu_{1}}\ \ \ \ \ (\ln
U\to+\infty),$

where we have made the change of variable $|c_{1}|^{2}=\eta$. It can be seen
in Eq. (24) that $c_{1}$ is a (complex) Gaussian random variable with $\langle
c_{1}\rangle=\langle c_{1}^{2}\rangle=0$ and
$\langle|c_{1}|^{2}\rangle=\mu_{1}$. Thus, writing $c_{1}=\sqrt{\eta}\,{\rm
e}^{i\arg(c_{1})}$ in Eq. (22), one gets

$S_{\rm inst}(z)=\sqrt{\eta}\,{\rm e}^{i\arg(c_{1})}\phi_{1}(z),$ (25)

where $\eta$ is an exponential random variable with $p(\eta)=\mu_{1}^{-1}{\rm
e}^{-\eta/\mu_{1}}$, and $\arg(c_{1})$ is a random phase uniformly distributed
over $[0,2\pi)$. For large $\ln U$ (hence large $\eta$), the realizations of
$S$ which contribute to the tail of $p(U)$ concentrate onto the leading
instanton, $S(z)\sim S_{\rm inst}(z)$ ($\eta\to+\infty$), and one recovers the
result of MD2004 ; MC2011  recalled in Eq. (3) (for $d_{1}=1$). It remains to
perform the integral over $\eta$ in Eq. (24), which can be done without
difficulty. One obtains the asymptotic behavior given in Eq. (I) with
$d_{1}=1$,

$\ln p(U)=-\left(1+\frac{1}{2\mu_{1}g}\right)\,\ln U+O(1)\ \ \ \ \ (\ln
U\to+\infty),$

as expected.

##  III Amplification with diffraction: general setting

The approach followed in the previous section to deal with the diffraction-
free case is completely different from the one in MD2004 ; MC2011 . Having
checked that both give the same results, we can now move on to the next step
and use the instanton analysis to deal with the full problem with diffraction.

###  III.1 MSR action and instanton equations

We consider the transversally one-dimensional ($d=1$) version of Eq. (1),

$\left\\{\begin{array}[]{l}\partial_{z}\psi(x,z)-\frac{i}{2m}\partial^{2}_{x^{2}}\psi(x,z)=g|S(x,z)|^{2}\psi(x,z),\\\
0\leq z\leq L,\ x\in\Lambda\subset\mathbb{R},\ {\rm and}\
\psi(x,0)=1,\end{array}\right.$ (26)

where we take for $\Lambda$ the circle of length $\ell$. The random field
$S(x,z)$ is homogeneous along $x$ with normalization
$L^{-1}\int_{0}^{L}\langle|S(x,z)|^{2}\rangle\,dz=1$. (The generalization to
more than one transverse dimension is straightforward.) Our goal is to
determine the realizations of $S(x,z)$ and the tail of $p(U)$ in the large
$\ln U$ limit for $U=|\psi(0,L)|^{2}$, with $\psi(x,z)$ solution to Eq. (26).
Write

$D_{z,\,x^{2}}\equiv\partial_{z}-\frac{i}{2m}\partial^{2}_{x^{2}},$ (27)

and $T_{C}$ the covariance operator of $S$ defined by

$(T_{C}f)(x,z)=\int_{\Lambda}\int_{0}^{L}C(x-x^{\prime},z,z^{\prime})\,f(x^{\prime},z^{\prime})\,dz^{\prime}\,dx^{\prime},\
\ \ f(x,z)\in L^{2}(\Lambda\times[0,L]),$ (28)

with $C(x-x^{\prime},z,z^{\prime})=\langle
S(x,z)S(x^{\prime},z^{\prime})^{\ast}\rangle$. The counterpart of Eq. (II.1)
in the problem with diffraction reads

$p(U)=\int_{\varphi(x,0)=1}\delta\left(U-|\varphi(0,L)|^{2}\right)\,{\rm
e}^{\frac{i}{2}\,\left(\left\langle\tilde{\varphi}\left|D_{z,\,x^{2}}-g|S|^{2}\right|\varphi\right\rangle+c.\,c.\right)-\left\langle
S\left|T_{C}^{-1}\right|S\right\rangle}\,\mathscr{D}^{2}\varphi\,\mathscr{D}^{2}\tilde{\varphi}\,\mathscr{D}^{2}S,$
(29)

which is of the same form as in Eq. (7) with MSR action

$\displaystyle\mathcal{A}(\varphi,\tilde{\varphi},S)=\frac{i}{2}\,\left(\left\langle\tilde{\varphi}\left|D_{z,\,x^{2}}-g|S|^{2}\right|\varphi\right\rangle+c.\,c.\right)-\left\langle
S\left|T_{C}^{-1}\right|S\right\rangle$
$\displaystyle=\frac{i}{2}\,\left[\int_{\Lambda}\int_{0}^{L}\tilde{\varphi}^{\ast}(x,z)\left(D_{z,\,x^{2}}\varphi(x,z)-g|S(x,z)|^{2}\varphi(x,z)\right)dz\,dx+c.\,c.\right]$
(30)
$\displaystyle-\int_{\Lambda}\int_{0}^{L}S^{\ast}(x,z)(T_{C}^{-1}S)(x,z)\,dz\,dx.$

The derivation of the instanton equations from the action in Eq. (III.1)
follows exactly the same line as in the diffraction-free case in Sec. II.2.
Varying $\mathcal{A}(\varphi,\tilde{\varphi},S)$ with the Lagrange multiplier
term $\lambda|\varphi(0,L)|^{2}$ and setting the variation to zero, one
obtains the equations

$\displaystyle\left[D_{z,\,x^{2}}-g|S(x,z)|^{2}\right]\varphi(x,z)=0\ {\rm
with}\ \varphi(x,0)=1,$
$\displaystyle\left[D_{z,\,x^{2}}+g|S(x,z)|^{2}\right]\tilde{\varphi}(x,z)=-2i\lambda\varphi(0,L)\delta(x)\delta(z-L)\
{\rm with}\ \tilde{\varphi}(x,L^{+})=0,$ (31)
$\displaystyle\left[D_{z,\,x^{2}}^{\ast}+g|S(x,z)|^{2}\right]\tilde{\vartheta}(x,z)=-2i\lambda\varphi^{\ast}(0,L)\delta(x)\delta(z-L)\
{\rm with}\ \tilde{\vartheta}(x,L^{+})=0,$

and

$(T_{C}^{-1}S)(x,z)=-\frac{ig}{2}\left[\tilde{\vartheta}(x,z)\varphi(x,z)+\tilde{\varphi}(x,z)\varphi^{\ast}(x,z)\right]S(x,z),$
(32)

or, equivalently,

$\displaystyle\left[D_{z,\,x^{2}}-g|S(x,z)|^{2}\right]\varphi(x,z)=0\ {\rm
with}\ \varphi(x,0)=1,$
$\displaystyle\left[D_{z,\,x^{2}}+g|S(x,z)|^{2}\right]\tilde{\varphi}(x,z)=0\
{\rm with}\ \tilde{\varphi}(x,L)=2i\lambda\varphi(0,L)\delta(x),$ (33)
$\displaystyle\left[D_{z,\,x^{2}}^{\ast}+g|S(x,z)|^{2}\right]\tilde{\vartheta}(x,z)=0\
{\rm with}\ \tilde{\vartheta}(x,L)=2i\lambda\varphi^{\ast}(0,L)\delta(x),$

and

$\left[T_{C}\left(\tilde{\vartheta}\varphi+\tilde{\varphi}\varphi^{\ast}\right)S\right](x,z)=\frac{2i}{g}S(x,z).$
(34)

The equations (III.1) are readily solved in terms of Feynman-Kac propagator,

$K(x_{2},z_{2};x_{1},z_{1})=\int_{x(z_{1})=x_{1}}^{x(z_{2})=x_{2}}{\rm
e}^{\int_{z_{1}}^{z_{2}}\left[\frac{im}{2}\dot{x}(\tau)^{2}+g|S(x(\tau),\tau)|^{2}\right]\,d\tau}\mathscr{D}x,$
(35)

with $z_{2}>z_{1}$, where the path-integral is over the set of all the
continuous paths in $\Lambda$ satisfying $x(z_{1})=x_{1}$ and
$x(z_{2})=x_{2}$. One gets

$\displaystyle\varphi(x,z)=\int_{\Lambda}K(x,z;y,0)\,dy,$
$\displaystyle\tilde{\varphi}(x,z)=2i\lambda\varphi(0,L)\,K(0,L;x,z)^{\ast},$
(36)
$\displaystyle\tilde{\vartheta}(x,z)=2i\lambda\varphi^{\ast}(0,L)\,K(0,L;x,z).$

Using the expressions (III.1) on the left-hand side of (34), one obtains

$\varphi^{\ast}(0,L)\,G_{1}(x,z)+\varphi(0,L)\,G_{2}(x,z)=\frac{1}{\lambda
g}\,S(x,z),$ (37)

with

$\displaystyle G_{1}(x,z)=$
$\displaystyle\int_{0}^{L}\int_{\Lambda}\int_{\Lambda}K(0,L;x^{\prime},z^{\prime})K(x^{\prime},z^{\prime};\xi,0)$
(38) $\displaystyle\times
C(x-x^{\prime},z,z^{\prime})\,S(x^{\prime},z^{\prime})\,dx^{\prime}\,d\xi\,dz^{\prime},$

and

$\displaystyle G_{2}(x,z)=$
$\displaystyle\int_{0}^{L}\int_{\Lambda}\int_{\Lambda}K(0,L;x^{\prime},z^{\prime})^{\ast}K(x^{\prime},z^{\prime};\xi,0)^{\ast}$
(39) $\displaystyle\times
C(x-x^{\prime},z,z^{\prime})\,S(x^{\prime},z^{\prime})\,dx^{\prime}\,d\xi\,dz^{\prime}.$

In the large $\ln U$ limit, $S$ concentrates onto the leading instanton,
$S_{\rm inst}$, solution to Eq. (37), and $\varphi(x,z)$ concentrates onto
$\varphi_{\rm inst}(x,z)$ given by the Feynman-Kac path-integral for
$\varphi(x,z)$ in Eq. (III.1) with $S=S_{\rm inst}$. For a given $C$ it is
always possible, in principle, to solve the instanton equation (37)
numerically. To this end, an iterative forward-backward scheme as introduced,
e.g., in CS2001  could be used to solve the equivalent system (III.1)-(34).
However, the fact that $\ln U$ is large allows Eq. (37) to be solved
analytically (in this limit) without the need to specify $C$. The key is to
call upon the well-known gain narrowing effect YL1963 , here in the space of
continuous paths $x(\cdot)$, according to which the Feynman-Kac propagators in
(38)-(39) are dominated by the contribution of the paths with the largest
amplification, the contribution of the other paths being subdominant in the
large amplification limit. These dominant trajectories run in the vicinity of
‘ridge paths’ along which $|S_{\rm inst}(x,z)|^{2}$ is at a global maximum for
every given $z$. Assuming that the ridge paths are all continuous (to be
checked a posteriori, once $S_{\rm inst}$ is known), Eq. (37) simplifies and
for a wide class of $S$ that we will now specify, it can be solved explicitly.

###  III.2 Specification of $\bm{S(x,z)}$

We assume that $S(x,z)$ can be expressed as a finite random Fourier sum,

$S(x,z)=\sum_{(n,j)\in\mathcal{I}}s_{(n,j)}\sqrt{\frac{\sigma_{(n,j)}}{\ell}}\,{\rm
e}^{2i\pi nx/\ell}\Phi_{(n,j)}(z),$ (40)

where $\mathcal{I}$ is a finite subset of $\mathbb{Z}\times\mathbb{N}$. The
$s_{(n,j)}$s are complex Gaussian random variables with $\langle
s_{(n,j)}\rangle=\langle s_{(n,j)}s_{(m,k)}\rangle=0$ and $\langle
s_{(n,j)}s_{(m,k)}^{\ast}\rangle=\delta_{nm}\delta_{jk}$, the
$\sigma_{(n,j)}$s are positive constants normalized to
$\sum_{(n,j)\in\mathcal{I}}\sigma_{(n,j)}=L\ell$, and, for fixed $n$, the
$\Phi_{(n,j)}$s are orthonormal continuous functions of $0\leq z\leq L$. Using
Eq. (40) in $C(x-x^{\prime},z,z^{\prime})=\langle
S(x,z)S(x^{\prime},z^{\prime})^{\ast}\rangle$ one gets

$C(x-x^{\prime},z,z^{\prime})=\sum_{(n,j)\in\mathcal{I}}\frac{\sigma_{(n,j)}}{\ell}\,{\rm
e}^{2i\pi
n(x-x^{\prime})/\ell}\Phi_{(n,j)}(z)\Phi_{(n,j)}(z^{\prime})^{\ast},$ (41)

from which it follows that $\sigma_{(n,j)}$ and ${\rm e}^{2i\pi
nx/\ell}\Phi_{(n,j)}(z)/\sqrt{\ell}$ are the eigenvalues and orthonormal
eigenfunctions of the covariance operator $T_{C}$ defined in Eq. (28).

Equation (40) generalizes models of spatially smoothed laser beams in which
laser light is represented by a superposition of monochromatic beamlets the
amplitudes of which are independent random variables RD1993 . For a large
number of beamlets these random variables can be taken as Gaussian and the
laser electric field takes on the form (40) in which the sum over $j$ reduces
to $j=1$ with $\Phi_{(n,1)}(z)=(1/\sqrt{L})\,\exp[i\alpha(2\pi n/\ell)^{2}z)]$
where $\alpha$ is a (real) constant. Moreover, every centered Gaussian field
with a continuous correlation function has an expansion of the form (40),
possibly with an infinite sum AT2007 . Combining this result with the
practically unavoidable existence of some natural cut-off making the sum
finite (like, e.g., in numerical simulations), one can safely expects the
expression in Eq. (40) to be quite generic, at least from a practical point of
view.

Let $B(0,L)$ denote the set of all the continuous paths in $\Lambda$
satisfying $x(L)=0$ and define $M[x(\cdot)]$ the
$|\mathcal{I}|\times|\mathcal{I}|$ positive definite matrix with components

$M_{(n,j)(m,k)}[x(\cdot)]=\frac{\sqrt{\sigma_{(n,j)}\sigma_{(m,k)}}}{\ell}\int_{0}^{L}{\rm
e}^{2i\pi(m-n)x(z)/\ell}\Phi_{(n,j)}(z)^{\ast}\Phi_{(m,k)}(z)\,dz,$ (42)

in which $x(\cdot)\in B(0,L)$. Since $M[x(\cdot)]$ is positive definite, all
its eigenvalues are real and positive. Write $\mu_{1}[x(\cdot)]>0$ the largest
eigenvalue of $M[x(\cdot)]$. It is proved in MCL2006  that the eigenvalues of
$M[x(\cdot)]$ are equal to the ones of $T_{x(\cdot)}$ defined by

$(T_{x(\cdot)}f)(z)=\int_{0}^{L}C(x(z)-x(z^{\prime}),z,z^{\prime})\,f(z^{\prime})\,dz^{\prime}.\
\ \ f(z)\in L^{2}([0,L]).$ (43)

It follows in particular that $\mu_{1}[x(\cdot)]$ is invariant under the path
transformations leaving $C(x(z)-x(z^{\prime}),z,z^{\prime})$ unchanged and
that the image of a path maximizing $\mu_{1}[x(\cdot)]$ by such a
transformation is also a path maximizing $\mu_{1}[x(\cdot)]$. We consider
cases fulfilling the following two assumptions:

(i) all the paths maximizing $\mu_{1}[x(\cdot)]$ are in $B(0,L)$;

(ii) there is a finite number of paths in $B(0,L)$ maximizing
$\mu_{1}[x(\cdot)]$.

Assumption (i) is a central feature of the class of $S$ we consider in this
paper. We don’t know whether Eq. (37) could be solved analytically in the
large $\ln U$ limit without this assumption. The technical restriction (ii)
will be used in Sec. IV.2. Lifting (ii) — e.g., in the case of an uncountable
set of maximizing paths — raises tricky technical problems yet to be solved;
this will be the subject of a future work. Both assumptions (i) and (ii) are
fulfilled in most cases of practical interest. Finally, for notational
convenience we define

$\mu_{\rm max}=\sup_{x(\cdot)\in B(0,L)}\mu_{1}[x(\cdot)],$ (44)

the supremum being reached in $B(0,L)$, by Assumption (i).

##  IV Single-filament instanton and tail of $\bm{p(U)}$

In the following, we consider the simplest case where for each realization of
$S_{\rm inst}$ there is only one ridge path of $|S_{\rm inst}(x,z)|^{2}$ in
$B(0,L)$, denoted by $x_{\rm inst}(\cdot)$, dominating the large $\ln U$ limit
of the Feynman-Kac propagators in (38) and (39). Note that $x_{\rm
inst}(\cdot)$ may be different from one realization of $S_{\rm inst}$ to the
other. The set of all the realizations of $S_{\rm inst}$ having the same
$x_{\rm inst}(\cdot)$ defines a random field, denoted by $S_{\rm inst}^{x_{\rm
inst}(\cdot)}$, referred to in the following as a ‘single-filament instanton’
(the reason for this name will appear more clearly at the end of Sec. IV.1).
We will write $\varphi_{\rm inst}^{x_{\rm inst}(\cdot)}(x,z)$ the Feynman-Kac
path-integral for $\varphi(x,z)$ in Eq. (III.1) with $S=S_{\rm inst}^{x_{\rm
inst}(\cdot)}$.

Single-filament instantons are not the only possible solutions to Eq. (37).
Multi-filament instantons are also possible if realizations of $|S_{\rm
inst}(x,z)|^{2}$ have more than one ridge path. The conditions for single- or
multi-filament instantons are specified below Eq. (49) as well as at the end
of Appendix A. The study of multi-filament instantons being excessively
intricate, we restrict ourselves to single-filament instantons for the sake of
clarity and readability.

###  IV.1 Leading instanton

Assume that $x_{\rm inst}(\cdot)$ is continuous (to be checked a posteriori).
As finite sums of continuous functions, both $S(x,z)$ and
$C(x-x^{\prime},z,z^{\prime})$ are continuous functions of their arguments. It
follows in particular that for fixed $x$, $z$, and $z^{\prime}$, the product
$C(x-x^{\prime},z,z^{\prime})\,S(x^{\prime},z^{\prime})$ on the right-hand
side of Eqs. (38) and (39) is a continuous function of $x^{\prime}$.
Integrating over $\xi$ and $x^{\prime}$ at fixed $z^{\prime}$ and using the
fact that, in the large $\ln U$ limit, only the vicinity of $x^{\prime}=x_{\rm
inst}(z^{\prime})$ contributes, one gets the large $\ln U$ behavior of
$G_{1,2}(x,z)$,

$\displaystyle
G_{1}(x,z)\sim\int_{\Lambda}K(0,L;\xi,0)\,d\xi\int_{0}^{L}C(x-x_{\rm
inst}(z^{\prime}),z,z^{\prime})\,S_{\rm inst}^{x_{\rm inst}(\cdot)}(x_{\rm
inst}(z^{\prime}),z^{\prime})\,dz^{\prime}$ (45) $\displaystyle=\varphi_{\rm
inst}^{x_{\rm inst}(\cdot)}(0,L)\,\int_{0}^{L}C(x-x_{\rm
inst}(z^{\prime}),z,z^{\prime})\,S_{\rm inst}^{x_{\rm inst}(\cdot)}(x_{\rm
inst}(z^{\prime}),z^{\prime})\,dz^{\prime}\ \ \ \ \ (\ln U\to+\infty),$

and

$\displaystyle
G_{2}(x,z)\sim\int_{\Lambda}K(0,L;\xi,0)^{\ast}\,d\xi\int_{0}^{L}C(x-x_{\rm
inst}(z^{\prime}),z,z^{\prime})\,S_{\rm inst}^{x_{\rm inst}(\cdot)}(x_{\rm
inst}(z^{\prime}),z^{\prime})\,dz^{\prime}$ (46)
$\displaystyle=\varphi(0,L)_{\rm inst}^{x_{\rm
inst}(\cdot)\,\ast}\,\int_{0}^{L}C(x-x_{\rm
inst}(z^{\prime}),z,z^{\prime})\,S_{\rm inst}^{x_{\rm inst}(\cdot)}(x_{\rm
inst}(z^{\prime}),z^{\prime})\,dz^{\prime}\ \ \ \ \ (\ln U\to+\infty).$

Injecting these expressions onto the left-hand side of Eq. (37), one obtains
the instanton equation

$\int_{0}^{L}C(x-x_{\rm inst}(z^{\prime}),z,z^{\prime})\,S_{\rm inst}^{x_{\rm
inst}(\cdot)}(x_{\rm
inst}(z^{\prime}),z^{\prime})\,dz^{\prime}=\frac{1}{2\lambda gU}\,S_{\rm
inst}^{x_{\rm inst}(\cdot)}(x,z),$ (47)

where we have used the equality $|\varphi_{\rm inst}^{x_{\rm
inst}(\cdot)}(0,L)|^{2}=U$ imposed by the delta function on the right-hand
side of (29).

Equation (47) can be solved in two different ways, depending on wether or not
the Fourier decompositions (40) and (41) for $S_{\rm inst}^{x_{\rm
inst}(\cdot)}$ and $C(x-x_{\rm inst}(z^{\prime}),z,z^{\prime})$ are used in
Eq. (47). Using these decompositions, one gets the eigenvalue equation

$\sum_{(m,k)\in\mathcal{I}}M_{(n,j)(m,k)}[x_{\rm
inst}(\cdot)]\,s_{(m,k)}=\frac{1}{2\lambda gU}\,s_{(n,j)},$ (48)

which fixes $\lambda$ at $\lambda=1/(2\mu_{n}[x_{\rm inst}(\cdot)]gU)$, where
$\mu_{1}[x_{\rm inst}(\cdot)]>\mu_{2}[x_{\rm inst}(\cdot)]>\cdots>0$ are the
eigenvalues of $M[x_{\rm inst}(\cdot)]$. Using (III.1) and (47) (or (48)) in
(III.1), one finds that the action of an instanton solution is
$\mathcal{A}=-\mu_{n}[x_{\rm inst}(\cdot)]^{-1}\|S(x_{\rm
inst}(\cdot),\cdot)\|_{2}^{2}$, and the leading instanton $S_{\rm
inst}^{x_{\rm inst}(\cdot)}$ for which $\mathcal{A}$ is maximum corresponds to
the largest eigenvalue $\mu_{1}[x_{\rm inst}(\cdot)]$ with $x_{\rm
inst}(\cdot)$ maximizing $\mu_{1}[x(\cdot)]$. Note that $x_{\rm inst}(\cdot)$
is a non-random path. The fact that $x_{\rm inst}(\cdot)$ exists and is
continuous is ensured by the assumption (i). Thus, for every path $x_{\rm
inst}(\cdot)\in B(0,L)$ maximizing $\mu_{1}[x(\cdot)]$, there is a leading
instanton

$S_{\rm inst}^{x_{\rm
inst}(\cdot)}(x,z)=\sum_{(n,j)\in\mathcal{I}}\mathfrak{s}_{(n,j)}\sqrt{\frac{\sigma_{(n,j)}}{\ell}}\,{\rm
e}^{2i\pi nx/\ell}\Phi_{(n,j)}(z),$ (49)

where $\bm{\mathfrak{s}}$ (with components $\mathfrak{s}_{(n,j)}$) is an
eigenvector of $M[x_{\rm inst}(\cdot)]$ associated with the largest eigenvalue
$\mu_{1}[x_{\rm inst}(\cdot)]$. It is checked in Appendix A that $x_{\rm
inst}(\cdot)$ is indeed a ridge path of $|S_{\rm inst}^{x_{\rm
inst}(\cdot)}(x,z)|^{2}$, as it should be.

The calculation in Appendix A also specifies under what condition a leading
instanton is a single-filament instanton. Namely, the fundamental eigenspace
of $M[x_{\rm inst}(\cdot)]$ and the one of $M[x(\cdot)]$ for every other path
maximizing $\mu_{1}[x(\cdot)]$, if any, must be essentially
disjoint111‘essentially disjoint’ and ‘trivial intersection’ mean that the
intersection reduces to the zero vector.. In particular, if the fundamental
eigenspaces of $M[x(\cdot)]$ for all the different paths maximizing
$\mu_{1}[x(\cdot)]$ are essentially disjoint, all the instantons are single-
filament instantons. This is the case considered in this paper. Conversely, if
the fundamental eigenspaces of $M[x(\cdot)]$ for different paths maximizing
$\mu_{1}[x(\cdot)]$ have a non trivial intersection, then multi-filament
instantons come into play as possible solutions to Eq. (37).

We now solve the equation (47) without using the Fourier decompositions (40)
and (41). It can be seen from Eq. (47) that $S_{\rm inst}^{x_{\rm
inst}(\cdot)}(x_{\rm inst}(z),z)$ is an eigenfunction of $T_{x_{\rm
inst}(\cdot)}$ with eigenvalue $1/2\lambda gU$, hence
$\lambda=1/(2\mu_{n}[x_{\rm inst}(\cdot)]gU)$, where $\mu_{1}[x_{\rm
inst}(\cdot)]>\mu_{2}[x_{\rm inst}(\cdot)]>\cdots>0$ are the eigenvalues of
$T_{x_{\rm inst}(\cdot)}$. (Recall that $T_{x(\cdot)}$ and $M[x(\cdot)]$ have
the same eigenvalues with the same multiplicities MCL2006 ). As explained
below Eq. (48), $S_{\rm inst}^{x_{\rm inst}(\cdot)}$ corresponds to the
largest eigenvalue $\mu_{1}[x_{\rm inst}(\cdot)]$ with $x_{\rm inst}(\cdot)$
maximizing $\mu_{1}[x(\cdot)]$. Like in Sec. II.2, we assume for simplicity
that $\mu_{\rm max}=\mu_{1}[x_{\rm inst}(\cdot)]$ is not degenerate. The case
of a degenerate $\mu_{\rm max}$ can be dealt with similarly without
difficulties (the calculation is more technical and the results not
substantially different). Writing $\phi_{1}$ the normalized fundamental
eigenmode of $T_{x_{\rm inst}(\cdot)}$, one has

$S_{\rm inst}^{x_{\rm inst}(\cdot)}(x_{\rm inst}(z),z)=c_{1}\phi_{1}(z),$ (50)

where $c_{1}$ is a complex number. Injecting (50) into (47), one obtains

$S_{\rm inst}^{x_{\rm inst}(\cdot)}(x,z)=\frac{c_{1}}{\mu_{\rm
max}}\,\int_{0}^{L}C(x-x_{\rm
inst}(z^{\prime}),z,z^{\prime})\,\phi_{1}(z^{\prime})\,dz^{\prime}.$ (51)

$\bullet$ Equivalence of the Fourier and convolution representations of
$S_{\rm inst}^{x_{\rm inst}(\cdot)}$

The fact that the expressions of $S_{\rm inst}^{x_{\rm inst}(\cdot)}(x,z)$ in
Eqs. (49) and (51) are equivalent is proved in Appendix B, with
$\mathfrak{s}_{(n,j)}$ in Eq. (49) and $c_{1}$ in Eq. (51) related to each
other by

$\mathfrak{s}_{(n,j)}=\frac{1}{\sqrt{\mu_{\rm
max}}}\,c_{1}\mathfrak{e}^{(1)}_{(n,j)}\ {\rm and}\ c_{1}=\sqrt{\mu_{\rm
max}}\,\sum_{(n,j)\in\mathcal{I}}\mathfrak{s}_{(n,j)}\mathfrak{e}^{(1)\,\ast}_{(n,j)},$
(52)

where $\bm{\mathfrak{e}}^{(1)}$ is the normalized fundamental eigenvector of
$M[x_{\rm inst}(\cdot)]$ defined by its components

$\mathfrak{e}^{(1)}_{(n,j)}=\sqrt{\frac{\sigma_{(n,j)}}{\ell\,\mu_{\rm
max}}}\,\int_{0}^{L}{\rm e}^{-2i\pi nx_{\rm
inst}(z^{\prime})/\ell}\Phi_{(n,j)}(z^{\prime})^{\ast}\phi_{1}(z^{\prime})\,dz^{\prime}.$
(53)

Note that by Eq. (49) for $S_{\rm inst}^{x_{\rm inst}(\cdot)}(x_{\rm
inst}(z),z)$ and the definition of $\mathfrak{e}^{(1)}_{(n,j)}$ in Eq. (53),
the expression of $c_{1}$ in Eq. (52) coincides with $c_{1}=\int_{0}^{L}S_{\rm
inst}^{x_{\rm inst}(\cdot)}(x_{\rm inst}(z),z)\phi_{1}(z)^{\ast}dz$ in
agreement with Eq. (50), as it should be.

$\bullet$ Statistical properties of $S$ in the large $\ln U$ limit

The statistical properties of $c_{1}$ and the $\mathfrak{s}_{(n,j)}$s are
readily obtained from the ones of the $s_{(n,j)}$s in Eq. (40). Since the
$s_{(n,j)}$s are i.i.d. standard complex Gaussian random variables with
$\langle s_{(n,j)}\rangle=\langle s_{(n,j)}^{2}\rangle=0$ and
$\langle|s_{(n,j)}|^{2}\rangle=1$, the orthogonal projection of the
$|\mathcal{I}|$-dimensional (complex) vector $\bm{s}$ with coordinates
$s_{(n,j)}$ onto any given direction is also a standard complex Gaussian
random variable statistically independent of the projections onto the
orthogonal directions. By (52), $c_{1}$ is proportional to the projection of
$\bm{s}$ onto the direction of the vector $\bm{\mathfrak{e}}^{(1)}$ defined in
(53), from which it follows that $c_{1}$ is a complex Gaussian random variable
with $\langle c_{1}\rangle=\langle c_{1}^{2}\rangle=0$ and
$\langle|c_{1}|^{2}\rangle=\mu_{\rm max}$. The statistical properties of the
$\mathfrak{s}_{(n,j)}$s are different from the ones of the $s_{(n,j)}$s
because $\bm{\mathfrak{s}}$ is restricted to the one-dimensional fundamental
eigenspace of $M[x_{\rm inst}(\cdot)]$, which induces correlations between the
$\mathfrak{s}_{(n,j)}$s. From the first Eq. (52) and the statistical
properties of $c_{1}$, one finds that the $\mathfrak{s}_{(n,j)}$s are
correlated complex Gaussian random variables with
$\langle\mathfrak{s}_{(n,j)}\rangle=\langle\mathfrak{s}_{(n,j)}\mathfrak{s}_{(m,k)}\rangle=0$
and
$\langle\mathfrak{s}_{(n,j)}\mathfrak{s}_{(m,k)}^{\ast}\rangle=\mathfrak{e}^{(1)}_{(n,j)}\mathfrak{e}^{(1)\,\ast}_{(m,k)}$.

Normalizing $S$ to its $L^{2}$-norm, Equation (51) and $S\sim S_{\rm
inst}^{x_{\rm inst}(\cdot)}$ ($\ln U\to+\infty$) yield

$\frac{S(x,z)}{\|S\|_{2}}\sim A{\rm e}^{i\arg(c_{1})}\int_{0}^{L}C(x-x_{\rm
inst}(z^{\prime}),z,z^{\prime})\,\phi_{1}(z^{\prime})\,dz^{\prime}\ \ \ \ \
(\ln U\to+\infty),$ (54)

where $\|\cdot\|_{2}$ denotes the $L^{2}$-norm over $\Lambda\times[0,L]$, $A$
is a positive constant, and $\arg(c_{1})$ is a random phase uniformly
distributed over $[0,2\pi)$. From Eq. (54) it follows immediately that
$|S(x,z)|/\|S\|_{2}$ is non-random, which means that the profile of $S(x,z)$
is purely deterministic in this case. This result generalizes the diffraction-
free deterministic profile of $S$ in Eq. (4) when diffraction is taken into
account.

$\bullet$ Typical shape of $S_{\rm inst}^{x_{\rm inst}(\cdot)}$

Although the Fourier representation of $S_{\rm inst}^{x_{\rm inst}(\cdot)}$ in
Eq. (49) is very useful to deal with technical points like in Appendix A, it
is not so clear as to the structure of $S_{\rm inst}^{x_{\rm inst}(\cdot)}$ in
real space. By contrast, it is easier to figure out the shape of $S_{\rm
inst}^{x_{\rm inst}(\cdot)}(x,z)$ from the convolution representation in Eq.
(51), knowing the correlation function $C(x-x^{\prime},z,z^{\prime})$. As a
simple illustration, take, e.g.,
$C(x-x^{\prime},z,z^{\prime})=f[(x-x^{\prime})/x_{c},(z-z^{\prime})/z_{c}]$,
where $x_{c}$ and $z_{c}$ respectively denote transverse and axial correlation
lengths, $f(x,z)$ being negligibly small outside the domain defined by both
$|x|\leq 1$ and $|z|\leq 1$. Assuming $x_{c}\ll\ell$, $z_{c}\ll L$ and a
‘gentle’ ridge path with $|\dot{x}_{\rm inst}(z)|\lesssim\ell/L$ for all
$0\leq z\leq L$, it is not difficult to show from Eq. (51) that $S_{\rm
inst}^{x_{\rm inst}(\cdot)}$ lives within a thin tube, or filament, of radius
$\rho\lesssim x_{c}+\ell z_{c}/L\ll\ell$ along the path $x_{\rm inst}(\cdot)$.
This is the reason for the name ‘single-filament instanton’ given to $S_{\rm
inst}^{x_{\rm inst}(\cdot)}$. An example of the elongated profile of $S_{\rm
inst}^{x_{\rm inst}(\cdot)}$ can be seen in Fig. 1 (Sec. V).

###  IV.2 Tail of $\bm{p(U)}$

Write $N_{\rm inst}$ the number of single-filament instantons. $N_{\rm inst}$
is the number of paths maximizing $\mu_{1}[x(\cdot)]$, which is finite by
assumption (ii). As mentioned below (49), we consider cases where the
fundamental eigenspaces of $M[x(\cdot)]$ for all the different paths
maximizing $\mu_{1}[x(\cdot)]$ are essentially disjoint. It means that the
instantons — that are all single-filament instantons — are mutually exclusive
realizations of $S$. As a result, the total instanton contribution to the tail
of $p(U)$ in Eq. (29) is the sum of the $N_{\rm inst}$ individual instanton
contributions. Let $\pi^{(i)}(U)$ denotes the contribution of the $i$th
instanton. It will be seen below that the leading term of $\ln\pi^{(i)}(U)$ in
the large $U$ limit does not depend on $i$. Writing $\ln f(U)$ this term and
$\pi^{(i)}(U)=f(U)A^{(i)}(U)$ with $\ln A^{(i)}(U)=o[\ln f(U)]$ as $\ln
U\to+\infty$, one has

$\displaystyle\ln p(U)$ $\displaystyle\sim$
$\displaystyle\ln\sum_{i=1}^{N_{\rm inst}}\pi^{(i)}(U)=\ln\sum_{i=1}^{N_{\rm
inst}}f(U)A^{(i)}(U)$ $\displaystyle=$ $\displaystyle\ln
f(U)+\ln\sum_{i=1}^{N_{\rm inst}}A^{(i)}(U)$ $\displaystyle=$
$\displaystyle\ln f(U)+o[\ln f(U)]\ \ \ \ \ (\ln U\to+\infty).$

Thus, at leading order, $\ln p(U)\sim\ln f(U)$ where $\ln f(U)$ is the leading
term of $\ln\pi^{(i)}(U)$ for all $1\leq i\leq N_{\rm inst}$. Let $[{\rm
LT}]^{(U\to+\infty)}(\cdot)$ denote the leading term of the asymptotic
expansion of $(\cdot)$ as $\ln U\to+\infty$. Picking a $i$ and integrating out
the fluctuations of the fields around the corresponding $i$th single-filament
instanton (with ridge-path $x_{\rm inst}(\cdot)$) in (29), one obtains

$\displaystyle\ln f(U)=[{\rm
LT}]^{(U\to+\infty)}\ln\left\langle\delta(U-|\varphi_{\rm inst}^{x_{\rm
inst}(\cdot)}(0,L)|^{2})\right\rangle_{S_{\rm inst}^{x_{\rm inst}(\cdot)}}$
(56) $\displaystyle=[{\rm
LT}]^{(U\to+\infty)}\ln\,\left[\int\delta\left(U-|\varphi_{\rm inst}^{x_{\rm
inst}(\cdot)}(0,L)|^{2}\right)\,{\rm e}^{-|c_{1}|^{2}/\mu_{\rm
max}}\,\frac{d^{2}c_{1}}{\pi\mu_{\rm max}}\right].$

To go further we need the behavior of $|\varphi_{\rm inst}^{x_{\rm
inst}(\cdot)}(0,L)|^{2}$ as a function of $c_{1}$ in the large $\ln U$ limit.
First, we make the change of variables $c_{1}=\sqrt{\eta}\,{\rm
e}^{i\arg(c_{1})}$, where $\eta$ is an exponential random variable with
$p(\eta)=\mu_{\rm max}^{-1}{\rm e}^{-\eta/\mu_{\rm max}}$, and $\arg(c_{1})$
is a random phase uniformly distributed over $[0,2\pi)$. For finite $L$ and
$\ell$, a large $\ln U$ implies a large $\eta$. It is clear from Eq. (51) and
the first Eq. (III.1) with $S=S_{\rm inst}^{x_{\rm inst}(\cdot)}$ that
$|\varphi_{\rm inst}^{x_{\rm inst}(\cdot)}(0,L)|^{2}$ does not depend on
$\arg(c_{1})$. Writing

$|\varphi_{\rm inst}^{x_{\rm inst}(\cdot)}(0,L)|^{2}=A(\eta)\,{\rm
e}^{2g\eta},$ (57)

without loss of generality, on the right-hand side of Eq. (56), one obtains

$\displaystyle\ln f(U)=[{\rm
LT}]^{(U\to+\infty)}\ln\int_{0}^{+\infty}\delta\left(U-A(\eta)\,{\rm
e}^{2g\eta}\right)\,p(\eta)\,d\eta$ $\displaystyle=[{\rm
LT}]^{(U\to+\infty)}\ln\int_{0}^{+\infty}\frac{\delta\left(\eta-\eta(U)\right)}{|2g+\partial_{\eta}\ln
A(\eta)|\,U}\,p(\eta)\,d\eta$ (58) $\displaystyle=[{\rm
LT}]^{(U\to+\infty)}\ln\frac{1}{U^{1+1/2\mu_{\rm
max}g}}\frac{\eta(U)\,A(\eta(U))^{1/2\mu_{\rm max}g}}{|2g+\partial_{\eta}\ln
A(\eta(U))|}$

where $\eta(U)$ is the solution to $A(\eta)\,{\rm e}^{2g\eta}=U$. It is shown
in Appendix C that

$\begin{array}[]{l}\lim_{\eta\to+\infty}\frac{1}{\eta}\,\ln A(\eta)=0,\\\
\lim_{\eta\to+\infty}\partial_{\eta}\ln A(\eta)=0,\end{array}$ (59)

from which it follows that $\eta(U)\sim(2g)^{-1}\ln U$ and
$|2g+\partial_{\eta}\ln A(\eta(U))|\sim 2g$ in Eq. (IV.2) which reads

$\displaystyle\ln f(U)$ $\displaystyle=$ $\displaystyle[{\rm
LT}]^{(U\to+\infty)}\ln\left(\frac{A((2g)^{-1}\ln U)^{1/2\mu_{\rm max}g}\,\ln
U}{4g^{2}\,U^{1+1/2\mu_{\rm max}g}}\right)$ (60) $\displaystyle=$
$\displaystyle-\left(1+\frac{1}{2\mu_{\rm max}g}\right)\,\ln U,$

where we have used $\ln A((2g)^{-1}\ln U)^{1/2\mu_{\rm max}g}=o(\ln U)$ (see
the end of appendix C). Note that the expression of $\ln f(U)$ in Eq. (60) is
independent of $i$, as announced. Using (60) on the right-hand side of (IV.2),
one finally obtains the tail of $p(U)$ as

$\ln p(U)=-\left(1+\frac{1}{2\mu_{\rm max}g}\right)\,\ln U+o(\ln U)\ \ \ \ \
(\ln U\to+\infty),$ (61)

from which it follows that $p(U)$ has a leading algebraic tail $\propto
U^{-\zeta}$ modulated by a slow varying amplitude (slower than algebraic) with
exponent $\zeta=(1+1/2\mu_{\rm max}g)$. Injecting this result into $\langle
U\rangle=\int_{1}^{+\infty}Up(U)\,dU$, one finds that the critical coupling in
the case of single-filament instantons is given by $g_{c}(L)=1/2\mu_{\rm
max}$, where $\mu_{\rm max}$ depends on $L$, in agreement with the general
result of Ref. MCL2006  recalled in Eq. (6).

##  V Numerical results

In this section we present the results of numerical simulations for
realizations of $S$ in a sample of experimentally realistic size. Of
particular interest are laser-plasma interaction experiments with space-time
optical smoothing in which $S$ is renewed periodically. For typical parameters
on current large laser facilities, the number of uncorrelated realizations of
$S$ generated after $1$ to $10$ shots of a $10$ ns laser beam with coherence
time between $0.5$ ps and $1$ ps is between $10^{4}$ and $2\ 10^{5}$, which
sets the size of the sample we consider here. Since the number of realizations
is limited, whether or not asymptotic results can be sampled is essential to
get an idea of what to expect — and not to expect — from the simulations. As a
Gaussian field, $S$ has a fast decreasing probability at large $|S|$ which
makes $p(\ln U\gg 1)$ so small that $\ln U\gg 1$ is out of reach of the sample
we consider. So we don’t expect to directly check the asymptotic instanton in
(51) match the numerical results, or the tail of $p(U)$ in (61) match the
numerical histogram of $U$. However, asymptotic behavior comes out gradually
as $U$ increases and instantons beginning to emerge from the noise may already
affect the results in the accessible, not asymptotic regime. If so,
simulations should show it and, hopefully, provide useful information on the
transition to the asymptotic regime.

One possible way to reach the asymptotic regime would be to bias the
underlying distribution of $S$ towards the outcomes of interest, like, e.g.,
in the ‘importance sampling algorithm’ HM1956  frequently used in rare event
physics (see e.g. HMS2019  and references therein). In principle, this
approach should allow the tail of $p(U)$ to be probed at high $\ln U\gg 1$. A
direct numerical test of our analytical results, which is important from a
theoretical viewpoint but of a somewhat lesser interest experimentally, will
be the subject of a future work.

For the sake of completeness, note that the comparison of asymptotic instanton
solutions to results of numerical simulations has been carried out for
nonlinear equations with additive noise in several works. See, e.g., GGS2013 ;
HMS2019  and references therein for the Burgers and Kardar-Parisi-Zhang
equations, respectively.

For definiteness, we have taken

$S(x,z)=\sum_{n=-50}^{50}s_{n}\sqrt{\varsigma_{n}}\,\exp\,i\left[\frac{2\pi
n}{\ell}x+\left(\frac{2\pi n}{\ell}\right)^{2}\frac{z}{2}\right],$ (62)

where the $s_{n}$s are complex Gaussian random variables with $\langle
s_{n}\rangle=\langle s_{n}s_{m}\rangle=0$ and $\langle
s_{n}s_{m}^{\ast}\rangle=\delta_{nm}$, and the spectral density
$\varsigma_{n}$ is normalized to $\sum_{n=-50}^{50}\varsigma_{n}=1$. Equation
(62) — which is of the form (40) in which the sum over $j$ reduces to $j=1$
and $\varsigma_{n}=\sigma_{(n,1)}/L\ell$ — is reminiscent of models of
spatially smoothed laser beams RD1993 , where $S$ is a solution to the
paraxial wave equation

$\partial_{z}S(x,z)+\frac{i}{2}\partial^{2}_{x^{2}}S(x,z)=0,$ (63)

here with boundary condition
$S(x,0)=\sum_{n=-50}^{50}s_{n}\sqrt{\varsigma_{n}}\,\exp(2i\pi nx/\ell)$. To
ensure that the space average $\ell^{-1}\int_{\Lambda}S(x,z)\,dx$ is zero for
all $z$ and every realization of $S$, as expected for the electric field of a
smoothed laser beam, the mode at $n=0$ is excluded by taking
$\varsigma_{0}=0$. Here we show the results for the Gaussian spectrum

$\varsigma_{n\neq 0}\propto\exp\left[-\left(\frac{\pi
n}{\ell}\right)^{2}\right].$ (64)

(Other widely used spectra, like top-hat and Cauchy spectra, give similar
results.)

For each realization of $S$ on a cylinder of length $L=10$ and circumference
$\ell=20$, we have solved Eq. (1) by using a symmetrized $z$-split method
Strang1968  which propagates the diffraction term,
$(i/2m)\partial^{2}_{x^{2}}\psi(x,z)$, in Fourier space and the amplification
term, $g|S(x,z)|^{2}\psi(x,z)$, in real space. We have taken $m=0.7$, and
$g=0.5$. To get a better statistics of large amplification values, we have
considered $U_{\rm max}=|\psi(x_{\rm max},L)|^{2}$ instead of
$U=|\psi(0,L)|^{2}$, where $x_{\rm max}$ is the value of $x$ maximizing
$|\psi(x,L)|^{2}$ (i.e., the location of the highest peak of
$|\psi(x,L)|^{2}$). As $\ln U_{\rm max}$ increases, $S$ is expected to
concentrate onto the leading instanton(s) arriving at $x(L)\simeq x_{\rm
max}$.

Write $S_{\rm inst}(x,z)$ the leading instanton(s) arriving at $x(L)=0$. For a
given $x(\cdot)$ arriving at $x(L)=0$ we can compute the elements of
$M[x(\cdot)]$, then its largest eigenvalue $\mu_{1}[x(\cdot)]$, numerically.
Maximizing the result over fractional polynomial paths numerically, we found a
unique global maximum at $x_{\rm inst}(\cdot)\equiv 0$ with non-degenerate
$\mu_{\rm max}=4.34984$. We have then determined $S_{\rm inst}$ from Eqs.
(49), (62), and (64), with numerically computed eigenvector
$\bm{\mathfrak{s}}$. The critical coupling is $g_{c}(L)=1/(2\mu_{\rm
max})\simeq~{}0.11495$ and $g=0.5$ is in the above critical regime with
$g/g_{c}(L)\simeq 4.35>1$. Figure 1 shows the contour plots of $|S_{\rm
inst}|^{2}$ and the ‘hot spot profile’ $|C|^{2}$ RD1993 ; G1999 .

Figure 1: (a) Contour plot of $|S_{\rm inst}(x,z)|^{2}$ normalized to
$\max_{\Lambda\times[0,L]}|S_{\rm inst}(x,z)|^{2}=1$. (b) Contour plot of the
‘hot spot profile’ $|C(x,z)|^{2}$.

By statistical invariance under $x$-translation, the leading instanton
arriving at $x(L)=y$ is $S_{\rm inst}^{y}(x,z)=S_{\rm inst}(x-y,z)$. Define
$\hat{S}_{\rm inst}^{y}=S_{\rm inst}^{y}/\|S_{\rm
inst}^{y}\|_{2,\Lambda\times[0,L]}$ and
$\hat{S}=S/\|S\|_{2,\Lambda\times[0,L]}$ where
$\|\cdot\|_{2,\Lambda\times[0,L]}$ is the $L^{2}$-norm on
$\Lambda\times[0,L]$. Write $S_{\parallel}^{y}=\left(\hat{S}_{\rm
inst}^{y},\hat{S}\right)\hat{S}_{\rm inst}^{y}$ the component of $\hat{S}$
along $S_{\rm inst}^{y}$. We have measured the difference between $S$ and the
instanton through the minimized $L^{2}$-distance

$\displaystyle\mathfrak{D}\equiv d_{2}\left(\hat{S},S_{\parallel}^{y_{\rm
min}}\right)$ $\displaystyle=$
$\displaystyle\min_{y\in\Lambda}\,\|\hat{S}-S_{\parallel}^{y}\|_{2,\Lambda\times[0,L]}$
(65) $\displaystyle=$
$\displaystyle\sqrt{1-\max_{y\in\Lambda}\,|(\hat{S},\hat{S}_{\rm
inst}^{y})|^{2}},$

where $y_{\rm min}$ is the value of $y$ minimizing
$\|\hat{S}-S_{\parallel}^{y}\|_{2,\Lambda\times[0,L]}$. The smaller
$\mathfrak{D}$, the closer $S$ to the instanton arriving at $x(L)=y_{\rm
min}$. The fact that $y_{\rm min}$ can be different from $x_{\rm max}$ is due
to the fluctuations of $S$ away from the instanton, the relative amplitude of
which is measured by $\mathfrak{D}$. For $\mathfrak{D}$ smaller than average,
$y_{\rm min}\simeq x_{\rm max}$ with a relatively small dispersion of the data
points about $y_{\rm min}=x_{\rm max}$, as can be seen in Fig. 5. Using the
Fourier representations (62) for both $S$ and $S^{y}_{\rm Inst}$ on the right-
hand side of Eq. (65), one gets

$\mathfrak{D}=\sqrt{1-\frac{\max_{y\in\Lambda}\,|\sum_{n}\varsigma_{n}\hat{s}_{n}\hat{\mathfrak{s}}_{n}^{\ast}{\rm
e}^{2i\pi
ny/\ell}|^{2}}{\left(\sum_{n}\varsigma_{n}|s_{n}|^{2}\right)\,\left(\sum_{n}\varsigma_{n}|\mathfrak{s}_{n}|^{2}\right)}},$
(66)

which is the expression we have used in the simulations.

Figure 2: Probability distribution of $\mathfrak{D}$ estimated from (a)
$\\{S\\}$, (b) and (c) the realizations in $\\{S\\}$ with $\ln U_{\rm max}$
above the $90$th and $99$th percentiles, respectively. Figure 3: Probability
distribution of $\log_{10}U_{\rm max}$ estimated from (a) $\\{S\\}$, (b)
$\\{S\\}_{10\%}$, and (c) $\\{S\\}_{1\%}$.

We drew $10^{5}$ independent realizations of $S$ denoted in the following by
$\\{S\\}$. Figure 2 shows the probability distribution of $\mathfrak{D}$
estimated from $\\{S\\}$ and the realizations in $\\{S\\}$ with $\ln U_{\rm
max}$ above the $90$th and $99$th percentiles. The last two are conditional
probabilities knowing that $\log_{10}U_{\rm max}\geq 12.7$ and
$\log_{10}U_{\rm max}\geq 18$, respectively. One can see a clear tendency of
$\mathfrak{D}$ to decrease with increasing $\log_{10}U_{\rm max}$: the
subsamples of $\\{S\\}$ conditioned on a larger $\log_{10}U_{\rm max}$ are
statistically biased toward the instanton compared with the unconditioned
sample $\\{S\\}$ itself. This numerical result is consistent with the
predicted concentration of $S$ onto the instanton for $\ln U\to+\infty$.

To study this bias in more detail, we have used the two samples
$\\{S\\}_{10\%}$ and $\\{S\\}_{1\%}$ respectively defined as the realizations
in $\\{S\\}$ with $\mathfrak{D}$ below the $10$th and $1$st percentiles. These
samples correspond to $\mathfrak{D}\leq 0.8$ for $\\{S\\}_{10\%}$ and
$\mathfrak{D}\leq 0.7$ for $\\{S\\}_{1\%}$. Figure 3 shows the probability
distribution of $\log_{10}U_{\rm max}$ estimated from (a) $\\{S\\}$, (b)
$\\{S\\}_{10\%}$, and (c) $\\{S\\}_{1\%}$. The last two are conditional
probabilities knowing that $\mathfrak{D}\leq 0.8$ and $\mathfrak{D}\leq 0.7$,
respectively. In this figure, the statistical bias of $S$, already observed in
Fig. 2, appears as the clear tendency of $\log_{10}U_{\rm max}$ to increase
with decreasing $\mathfrak{D}$.

Figure 4: Percentage of realizations of $S$ in $\\{S\\}_{10\%}$ (orange, upper
curve) and $\\{S\\}_{1\%}$ (green, lower curve) for $\log_{10}U_{\rm max}$ in
$[n,\,n+2)$, with $n$ varying from $0$ to $28$. As guides to the eyes, the
solid lines are nonlinear fits of the corresponding data points. Dashed lines
are continuations of these fits to higher $\log_{10}U_{\rm max}$ (disregarding
the data points in this domain)

The concentration of $S$ onto the instanton implies that for all
$\varepsilon>0$ and $0<a<b$, one has $\lim_{a\to+\infty}{\rm
Prob.}(\mathfrak{D}\leq\varepsilon|\,a\leq\log_{10}U_{\rm max}<b)=1$. Thus,
for $a$ large enough it is not unreasonable to expect ${\rm
Prob.}(\mathfrak{D}\leq\varepsilon|\,a\leq\log_{10}U_{\rm max}<b)$ to increase
with increasing $a$, which should be possible to check numerically. ${\rm
Prob.}(\mathfrak{D}\leq\varepsilon|\,a\leq\log_{10}U_{\rm max}<b)$ can be
estimated by the percentage of realizations of $S$ with
$\mathfrak{D}\leq\varepsilon$ among the realizations with
$a\leq\log_{10}U_{\rm max}<b$. In Figure 4 we show the results for
$\varepsilon=0.8$ and $0.7$ (i.e., $S$ in $\\{S\\}_{10\%}$ and
$\\{S\\}_{1\%}$, respectively), $a=n$, and $b=n+2$, with $n$ an integer. It
can be seen that both curves increase with increasing $\log_{10}U_{\rm max}$,
as expected. Note, e.g., that $30\%$ of the realizations with $\log_{10}U_{\rm
max}\simeq 28$ are in $\\{S\\}_{1\%}$ (i.e., have $\mathfrak{D}\leq 0.7$),
when $\\{S\\}_{1\%}$ represents only $1\%$ of all the realizations in
$\\{S\\}$: the emergence of a statistical bias of $S$ with increasing
amplification is clearly visible.

Figure 5: Scatter plot of $x_{\rm max}$ and $y_{\rm min}$ for the realizations
in $\\{S\\}$ with $\log_{10}U_{\rm max}\geq 18$. Red circles and gray squares
are realizations in $\\{S\\}_{1\%}$ ($\mathfrak{D}\leq 0.7$) and
$\\{S\\}\setminus\\{S\\}_{1\%}$ ($\mathfrak{D}>0.7$), respectively.

Figure 5 shows a scatter plot of $x_{\rm max}$ and $y_{\rm min}$ for the
$10^{3}$ realizations in $\\{S\\}$ with the largest $U_{\rm max}$ (viz.,
$\log_{10}U_{\rm max}\geq 18$). Red circles and gray squares correspond to
realizations with $\mathfrak{D}\leq 0.7$ and $\mathfrak{D}>0.7$, respectively
(i.e., realizations in $\\{S\\}_{1\%}$ and $\\{S\\}\setminus\\{S\\}_{1\%}$).
It can be checked that the dispersion of the data points about $y_{\rm
min}=x_{\rm max}$ is indeed smaller for smaller $\mathfrak{D}$, as announced
below Eq. (65). Note that due to the periodic boundary condition in $\Lambda$,
the distance between $x_{\rm max}$ and $y_{\rm min}$ is $\min(|x_{\rm
max}-y_{\rm min}|,\ell-|x_{\rm max}-y_{\rm min}|)$ and the data points in the
left-upper and right-lower corners are actually close to the diagonal.

Figure 6: Contour plots of $|S(x,z)|^{2}$ for two realizations in
$\\{S\\}_{1\%}$ ($\mathfrak{D}\leq 0.7$) with $\log_{10}U_{\rm max}\geq 27$.
The dashed contours indicate the theoretical instanton arriving at
$x(L)=y_{\rm min}$ for the considered realization. (a): $\log_{10}U_{\rm
max}=29.63$, $\mathfrak{D}=0.68$, and
$\max_{\Lambda\times[0,L]}|S(x,z)|^{2}=14.83$. (b): $\log_{10}U_{\rm max}=27$,
$\mathfrak{D}=0.537$, and $\max_{\Lambda\times[0,L]}|S(x,z)|^{2}=13$. Figure
7: Plots similar to the ones shown in Figure 6 for two realizations in
$\\{S\\}\setminus\\{S\\}_{1\%}$ ($\mathfrak{D}>0.7$). (a): $\log_{10}U_{\rm
max}=29.84$, $\mathfrak{D}=0.8$, and
$\max_{\Lambda\times[0,L]}|S(x,z)|^{2}=17.79$. (b): $\log_{10}U_{\rm
max}=27.69$, $\mathfrak{D}=0.82$, and
$\max_{\Lambda\times[0,L]}|S(x,z)|^{2}=17.12$.

We have compared the realizations in $\\{S\\}_{1\%}$ and
$\\{S\\}\setminus\\{S\\}_{1\%}$ near the edge of the sampled domain of $U_{\rm
max}$, where $\\{S\\}_{1\%}$ becomes statistically significant according to
the results in Fig. 4. We have considered realizations with $\log_{10}U_{\rm
max}\geq 27$. There are $15$ such realizations in $\\{S\\}$ among which $5$ in
$\\{S\\}_{1\%}$ ($\mathfrak{D}\leq 0.7$) and $10$ in
$\\{S\\}\setminus\\{S\\}_{1\%}$ ($\mathfrak{D}>0.7$). In Figures 6 and 7 we
show two pairs of typical realizations picked in $\\{S\\}_{1\%}$ and
$\\{S\\}\setminus\\{S\\}_{1\%}$, respectively (technical details are given in
the captions). For each realization, the theoretical instanton arriving at
$x(L)=y_{\rm min}$ is indicated by a dashed contour, solution to $|S_{\rm
inst}(x-y_{\rm min},z)|^{2}=0.75$ with $|S_{\rm inst}|^{2}$ as in Fig. 1.
Intense localized hot spots similar to the theoretical one in Fig. 1(b) are
clearly visible in both figures. In Fig. 6 ($\mathfrak{D}\leq 0.7$), hot spots
occur inside the dashed line, in the instanton region. Note also that the
level of $|S(x,z)|^{2}$ is significantly higher than average throughout the
instanton region ($\sim 6$, while $\langle|S(x,z)|^{2}\rangle=1$), which seems
difficult to explain by generic fluctuations (i.e. independent, small-scale
hot spots). On the other hand, in Fig. 7 ($\mathfrak{D}>0.7$), hot spots occur
anywhere and the levels of $|S(x,z)|^{2}$ inside and outside the instanton
region are quite comparable (hot spots excluded).

The robustness of these observations from one realization to the other can be
tested through the sample mean of $|\hat{S}(x+x_{\rm max},z)|^{2}$ in which
the realizations are translated to align the maxima of $|\psi(x,L)|^{2}$ with
each other at the same position (here, $x=0$). In Figure 8, we show the
results for the same $15$ realizations with $\log_{10}U_{\rm max}\geq 27$ as
above. The region of the sample mean of $|\hat{S}_{\rm inst}(x+x_{\rm
max}-y_{\rm min},z)|^{2}$ is indicated by a dashed contour.

Figure 8: Contour plots of the sample mean of $|\hat{S}(x+x_{\rm max},z)|^{2}$
for the realizations in (a) $\\{S\\}_{1\%}$ and (b)
$\\{S\\}\setminus\\{S\\}_{1\%}$, with $\log_{10}U_{\rm max}\geq 27$. The
dashed contours indicate the region where the sample mean of $|\hat{S}_{\rm
inst}(x+x_{\rm max}-y_{\rm min},z)|^{2}$ is greater than $75\%$ of its maximum
value.

The overintensity in both figures (a) and (b) at $x=0$ and $z\simeq L$ is an
effect of diffraction characteristic of the sub-asymptotic regime
($\log_{10}U_{\rm max}\not\gg 1$), as we will now explain. Consider first a
generic realization of $S$ with $\log_{10}U_{\rm max}$ in the bulk of
$p(\log_{10}U_{\rm max})$. In this case, $S$ is a collection of hot spots
Dixit1993 ; G1985 ; RD1993  — or a ‘hot spot field’ — with no visible
instanton. Assume for the sake of argument that there is only one hot spot.
For a given $U_{\rm max}$ the hot spot is either right behind the maximum of
$|\psi(x,L)|^{2}$ (looking from $z>L$), or away from it with a larger
amplitude to compensate for the diffraction loss. The most probable
configuration is a compromise between those two options, and the fast
decreasing probability of the hot spot intensity tilts the compromise in favor
of the former. Namely, the hot spot is more likely to find itself right behind
the maximum of $|\psi(x,L)|^{2}$. Generalizing to several hot spots, the same
reasoning leads to the same conclusion: a generic realization of $S$ is more
likely to have one hot spot right behind the maximum of $|\psi(x,L)|^{2}$ than
not. This bias is cumulative in the calculation of the sample mean of
$|\hat{S}(x+x_{\rm max},z)|^{2}$ which, as a result, is maximum at $x=0$ and
$z\simeq L$. The situation is completely different if diffraction is switched
off ($m^{-1}=0$). In this case, only the amplification along the straight path
$x(z)=x_{\rm max}$ contributes to the maximum of $|\psi(x,L)|^{2}$, without
diffraction loss, and the contributing hot spots can be anywhere along this
path. This is exemplified in Figs. 9(a) and (b) that show the sample mean of
$|\hat{S}(x+x_{\rm max},z)|^{2}$ for $200$ generic realizations of $S$ with
$\log_{10}U_{\rm max}\leq 10$ in the bulk of $p(\log_{10}U_{\rm max})$ (see
Fig. 3), with and without diffraction, respectively.

Figure 9: Contour plots of the sample mean of $|\hat{S}(x+x_{\rm max},z)|^{2}$
for $200$ generic realizations of $S$ with $\log_{10}U_{\rm max}\leq 10$ in
the bulk of $p(\log_{10}U_{\rm max})$, (a) with diffraction ($m=0.7$) and (b)
without diffraction ($m=+\infty$).

The results for $\log_{10}U_{\rm max}\geq 27$ in Fig. 8 are similar to those
in Fig. 9(a) except that the instanton begins to emerge from the hot spot
background in a non-negligible fraction of realizations ($\sim 30\%$), as seen
in Figs. 4, 6 and 8(a). In conclusion, the overintensity in both Figs. 8(a)
and (b) at $x=0$ and $z\simeq L$ is an effect of the diffraction induced bias
of the hot spot field, unrelated to the asymptotic instanton solutions. We
refer the interested reader to the diffraction-free results for the sample
mean of $|S|^{2}$ in MD2004  that show the emergence of the instanton but no
overintensity near $z=L$, in agreement with our analysis. Lastly, the
realizations of $S$ in the asymptotic limit $\ln U_{\rm max}\to+\infty$ are
instanton dominated with negligible hot spot contribution to $|\psi(x,L)|^{2}$
near its maximum. So, in this limit, it makes no difference whether or not
there is a hot spot right behind the maximum of $|\psi(x,L)|^{2}$ and there
should be no diffraction induced bias of the (subdominant) hot spot field in
the asymptotic regime.

Figure 10: Positions of the local maxima of $|S(x,z)|^{2}$ higher than $75\%$
of the global maximum (blue circles) for the same realizations as in Fig. 6.
High maxima cluster in the instanton region indicated by the dashed contour.
(See caption of Fig. 6 for details.)

We now turn to the rest of Fig. 8. Figure 8(a) shows the result for the
realizations in $\\{S\\}_{1\%}$. In substance, it confirms the observations
already made about the Fig. 6; namely, the observed level of $|S(x,z)|^{2}$
inside the dashed contour is the superposition of an average elevation of the
level (the emerging instanton) and fluctuations of comparable amplitude. The
presence of such an average elevation inside the dashed contour increases the
probability that high maxima of $|S(x,z)|^{2}$ occur inside the instanton
region. It is a pure statistical effect similar to the well known enhancement
of correlations of peaks in Gaussian fields Kaiser1984 ; BBKS1986 , the large-
scale instanton playing the same role as the ‘signal’ and ‘background field’
in Kaiser1984  and BBKS1986 , respectively. As a consequence, one observes (i)
a tendency for the hot spots to cluster in the instanton region and (ii) a
level of $|S(x,z)|^{2}$ between the hot spots significantly higher than the
average level outside the instanton region. These two points (i) and (ii)
signal the emergence of the instanton in the realizations of $\\{S\\}_{1\%}$.
Hot spot clustering is illustrated in Fig. 10 for the same realizations as in
Fig. 6. By contrast, no particular structure is observed in Fig. 8(b) for the
realizations in $\\{S\\}\setminus\\{S\\}_{1\%}$ (except the overintensity at
$x=0$ and $z\simeq L$). It means that neither emerging instanton nor
clustering of hot spots are significant in those realizations.

Combining numerical results with analytical predictions, we can now infer how
the transition to the asymptotic regime occurs as $\ln U$ increases. As long
as the value of $\ln U$ is in the bulk of $p(\ln U)=Up(U)$, the overwhelming
majority of the realizations of $S$ are generic realizations with hot spots
uniformly scattered in $\Lambda\times[0,L]$ and $\mathfrak{D}$ close to its
typical value at $\mathfrak{D}\simeq 0.86$. The situation changes gradually as
$\ln U$ increases into the tail of $p(\ln U)$, as seen in Fig. 4. Namely, the
larger $\ln U$ the larger the percentage of atypical realizations with
$\mathfrak{D}$ smaller than, say, its first percentile — like the ones in Fig.
6 — to the detriment of generic realizations — like the ones in Fig. 7. In
those atypical realizations, the hot spots cluster in the instanton region
instead of being uniformly scattered in $\Lambda\times[0,L]$ and the level of
$|S(x,z)|^{2}$ between the hot spots remains abnormally high (see Figs. 6,
8(a), and 10). Letting $\ln U\to+\infty$, the percentage of atypical
realizations goes up to $100\%$ while $\mathfrak{D}$ and the relative
fluctuations-to-instanton amplitude decrease to zero with probability one. In
this limit, the tail of $p(U)$ is asymptotically dominated by the instanton
which determines the the critical coupling $g_{c}(L)$.

##  VI Discussion and perspectives

In this paper, we have studied the large amplification limit of a linear
amplifier driven by the square of a Gaussian random field. We have considered
the same model as in Refs. RD1994  and MCL2006  in which the propagation is
that of a free Schrödinger equation. By performing the first instanton
analysis of the corresponding MSR action, we have identified the realizations
of the Gaussian field most likely to produce a large amplification. We have
found that when $\ln U=\ln|\psi(0,L)|^{2}$ gets large, for $\psi$ solution to
Eq. (1) with $S$ defined in Sec. III.2, the realizations of $S$ concentrate
onto large-scale filamentary instantons running along the path(s) maximizing
the largest eigenvalue of the covariance operator defined in Eq. (43). This
result explains the otherwise mysterious presence of this maximized eigenvalue
in the expression of $g_{c}(L)$ found in MCL2006  (see Eq. (6)). We have then
derived the tail of $p(U)$ for large $\ln U$ from the instanton contribution
and checked that the resulting critical coupling does coincide with the one in
Ref. MCL2006 . From this analysis, it follows in particular that the
realizations of $S$ causing the divergence of $\langle|\psi(0,L)|^{2}\rangle$
for $g>g_{c}(L)$ are long filamentary structures (the instantons) rather than
localized hot-spots, as assumed in hot-spot models RD1994 . This result
extends the conclusions of Ref. MD2004  to the full problem (1) with
diffraction.

Numerical simulations clearly show a statistical bias of $S$ towards the
instanton, as $\ln U$ increases. The larger $\ln U$ in the sampled range, the
larger the fraction of atypical realizations of $S$ in which a large-scale
instanton coexists with fluctuation induced localized hot spots. (See MD2004
for a quantitative comparison of hot spot and instanton contributions to the
amplification in the diffraction-free case.) In those atypical realizations,
hot spots are not uniformly distributed in $\Lambda\times[0,L]$ but tend to
cluster in the instanton region. For the experimentally realistic sample size
we considered ($\sim 10^{5}$), it proved impossible to probe values of $\ln U$
large enough that the fluctuations of $S$ away from the instanton could be
neglected. Hot spot clustering and nonlinear evolution of the coupled hot
spots/instanton system are interesting subjects that would deserve to be dealt
with in more depth, especially in laser-plasma interaction physics.

The work presented here is only a first step toward a comprehensive study of
Eq. (1) in the large amplification limit. There are various directions along
which investigations could be pushed further. Obviously, trying to lift all or
part of the assumptions made in Secs. III.2 and IV appears as a natural next
step, especially the technical restriction (ii) and the possibility of multi-
filament instantons. As mentioned at the beginning of Sec. V, it would also be
important to directly test the validity of our analytical results, either by
significantly increasing the sample size, or by using a biased numerical
scheme capable of probing $p(U)$ in the asymptotic regime.

Another challenging line of research is the study of a possible intermittency
of $|\psi(x,L)|^{2}$ and its connection with our results, as we will now
explain. The experimental conditions to which our results can be directly
applied are those that naturally sample the realizations of $S$, like, e.g.,
in a laser-plasma interaction experiment with space-time optical smoothing in
which $S$ is renewed periodically. The situation is different in the case of
purely spatial smoothing, where a unique realization of $S$ is available in a
given experimental environment and $\langle|\psi(0,L)|^{2}\rangle$ is replaced
with the space average $|\Lambda|^{-1}\int_{\Lambda}|\psi(x,L)|^{2}\,dx$ for a
generic realization of $S$. As rare events, instantons are very unlikely to
contribute to the latter quantity unless $|\Lambda|$ is large and the space
average is dominated by the contribution of scarce, intense peaks of
$|\psi(x,L)|^{2}$ the high amplitude of which outbalances their scarcity. The
question is then whether such a peak-dominated behavior — called
‘intermittency’ in the literature on random media Molchanov1991  — can be
observed in the solution to Eq. (1) for large $|\Lambda|$ and $g>g_{c}(L)$. If
so, our results imply that $S(x,z)$ in the region of $\Lambda\times[0,L]$
upstream from a dominant peak of $|\psi(x,L)|^{2}$ is a filament instanton
arriving at the peak location. Intermittency of $|\psi(x,L)|^{2}$ is thus
important as connecting our instanton analysis approach with experimental
results for a given realization of $S$ in the large $|\Lambda|$ limit and
$g>g_{c}(L)$. The interested reader will find a detailed introduction to
intermittency in random media in Ref. Molchanov1991 .

Finally, it would also be interesting to investigate the small $m$ behavior of
the same problem. For $m\to 0$, it can be shown that $\psi(x,z)$ reduces to

$\psi(x,z)=\exp\left(\frac{g}{|\Lambda|}\,\|S\|^{2}_{2,\Lambda\times[0,L]}\right).$
(67)

Thus, in this limit, the realizations of $S$ giving rise to a large $\ln U$
are the ones with a large $L^{2}$-norm, which are known to concentrate onto
the fundamental eigenspace of the covariance operator $T_{C}$ defined in Eq.
(28), as $\ln U\to+\infty$ MD2004 ; MC2011 . If $S$ is given by the random
Fourier sum (40) with, e.g., $\sigma_{(n,j)}<\sigma_{(0,0)}$ for all
$(n,j)\neq(0,0)$, the fundamental eigenspace of $T_{C}$ reduces to the
functions $\propto\Phi_{(0,0)}(z)$ independent of $x$, and the realizations of
$S$ in the large $\ln U$ limit are completely delocalized in $\Lambda$, in
striking contrast to the filamentary instantons we have found for a fixed
$m\neq 0$. This simple example indicates that the two limits $\ln U\to+\infty$
and $m\to 0$ do not commute, which raises the natural question of how
precisely the crossover between ‘$\ln U\to+\infty$ then $m\to 0$’ and ‘$m\to
0$ then $\ln U\to+\infty$’ occurs. Answering this question will elucidate the
intriguing transition suggested by the above example, from filamentary to
delocalized instantons, as $m$ goes to zero.

In conclusion, it may be noted that the number of highly non-trivial questions
raised by the seemingly simple linear problem (1) is quite remarkable.
Following on from the work presented here, we hope that those questions will
motivate interesting research in both statistical physics and laser-matter
interaction physics where the linear amplifier model (1) first appeared.

###### Acknowledgements.

The author warmly thanks Satya N Majumdar, Denis Pesme, and Grégory Schehr for
their interest and valuable advice about the manuscript. He also thanks Harvey
A Rose and Joel L Lebowitz for the inspiring discussions he had with them on
related subjects.

##  Appendix A Paths maximizing $\bm{\mu_{1}[x(\cdot)]}$ and ridge paths of
$\bm{|S_{\rm inst}(x,z)|^{2}}$

In this appendix we show that $x_{\rm inst}(\cdot)$ is a ridge path of
$|S_{\rm inst}^{x_{\rm inst}(\cdot)}(x,z)|^{2}$. From Eqs. (42) and (49), one
gets

$\int_{0}^{L}|S_{\rm inst}^{x_{\rm inst}(\cdot)}(x_{\rm
inst}(z),z)|^{2}\,dz=\bm{\mathfrak{s}}^{\dagger}M[x_{\rm
inst}(\cdot)]\bm{\mathfrak{s}}=\mu_{\rm max}\|\bm{\mathfrak{s}}\|^{2},$ (68)

where $\bm{\mathfrak{s}}$ is in the fundamental eigenspace of $M[x_{\rm
inst}(\cdot)]$, and

$\displaystyle\int_{0}^{L}|S_{\rm inst}^{x_{\rm
inst}(\cdot)}(x(z),z)|^{2}\,dz=\bm{\mathfrak{s}}^{\dagger}M[x(\cdot)]\bm{\mathfrak{s}}\leq\mu_{1}[x(\cdot)]\|\bm{\mathfrak{s}}\|^{2}$
$\displaystyle=\frac{\mu_{1}[x(\cdot)]}{\mu_{\rm max}}\int_{0}^{L}|S_{\rm
inst}^{x_{\rm inst}(\cdot)}(x_{\rm inst}(z),z)|^{2}\,dz\leq\int_{0}^{L}|S_{\rm
inst}^{x_{\rm inst}(\cdot)}(x_{\rm inst}(z),z)|^{2}\,dz,$ (69)

yielding

$\sup_{x(\cdot)\in B(0,L)}\int_{0}^{L}|S_{\rm inst}^{x_{\rm
inst}(\cdot)}(x(z),z)|^{2}\,dz=\int_{0}^{L}|S_{\rm inst}^{x_{\rm
inst}(\cdot)}(x_{\rm inst}(z),z)|^{2}\,dz.$ (70)

Equation (70) means that in the path-integral for $\varphi_{\rm inst}^{x_{\rm
inst}(\cdot)}(0,L)$, $x_{\rm inst}(\cdot)$ is a path along which the
amplification is maximum. Now, assume that there is $A\subset[0,L]$ with
$|A|\equiv\int_{0}^{L}\bm{1}_{z\in A}dz>0$ such that for all $z\in A$, there
is $x\in\Lambda$ with $|S_{\rm inst}^{x_{\rm inst}(\cdot)}(x,z)|^{2}>|S_{\rm
inst}^{x_{\rm inst}(\cdot)}(x_{\rm inst}(z),z)|^{2}$. It follows immediately
that

$\int_{0}^{L}\sup_{x\in\Lambda}|S_{\rm inst}^{x_{\rm
inst}(\cdot)}(x,z)|^{2}dz>\int_{0}^{L}|S_{\rm inst}^{x_{\rm
inst}(\cdot)}(x_{\rm inst}(z),z)|^{2}\,dz,$ (71)

and from Eqs. (70) and (71) one should have

$\int_{0}^{L}\sup_{x\in\Lambda}|S_{\rm inst}^{x_{\rm
inst}(\cdot)}(x,z)|^{2}dz>\sup_{x(\cdot)\in B(0,L)}\int_{0}^{L}|S_{\rm
inst}^{x_{\rm inst}(\cdot)}(x(z),z)|^{2}\,dz,$ (72)

in contradiction with the Lemma A1 in Ref. MCL2006  according to which one
must have an equality. Thus, there is no such $A$ and since every given
realization of $S_{\rm inst}^{x_{\rm inst}(\cdot)}(x,z)$ in Eq. (49) is a
continuous function of $x$ and $z$, one has $|S_{\rm inst}^{x_{\rm
inst}(\cdot)}(x,z)|^{2}\leq|S_{\rm inst}^{x_{\rm inst}(\cdot)}(x_{\rm
inst}(z),z)|^{2}$ for all $0\leq z\leq L$ and $x\in\Lambda$. This proves that
for all the realizations of $S_{\rm inst}^{x_{\rm inst}(\cdot)}(x,z)$ in Eq.
(49) with $\bm{\mathfrak{s}}$ in the fundamental eigenspace of $M[x_{\rm
inst}(\cdot)]$, $x_{\rm inst}(\cdot)$ is a ridge path of $|S_{\rm
inst}^{x_{\rm inst}(\cdot)}(x,z)|^{2}$ along which the amplification is
maximum.

Assume that there is a $\bm{\mathfrak{s}}$ in the fundamental eigenspace of
$M[x_{\rm inst}(\cdot)]$ and $y_{\rm inst}(\cdot)\in B(0,L)$ with $y_{\rm
inst}(\cdot)\neq x_{\rm inst}(\cdot)$ such that $y_{\rm inst}(\cdot)$ is also
a ridge path of $|S_{\rm inst}^{x_{\rm inst}(\cdot)}(x,z)|^{2}$ along which
the amplification is maximum. Then, $y_{\rm inst}(\cdot)$ maximizes
$\mu_{1}[x(\cdot)]$ and $\bm{\mathfrak{s}}$ belongs to the fundamental
eigenspace of $M[y_{\rm inst}(\cdot)]$ (otherwise, the amplification along
$y_{\rm inst}(\cdot)$ would be less than along $x_{\rm inst}(\cdot)$). Since
$\bm{\mathfrak{s}}$ belongs to the fundamental eigenspaces of both $M[x_{\rm
inst}(\cdot)]$ and $M[y_{\rm inst}(\cdot)]$, their intersection is necessarily
non trivial. It shows that the number of ridge paths depends on the relative
structure of the fundamental eigenspaces of $M[x(\cdot)]$ for the different
paths maximizing $\mu_{1}[x(\cdot)]$. If the fundamental eigenspaces of
$M[x(\cdot)]$ for all the paths maximizing $\mu_{1}[x(\cdot)]$ are essentially
disjoint, $\bm{\mathfrak{s}}$ cannot belong to more than one fundamental
eigenspace and each realization of the instanton has only one ridge path. This
is the case considered in the paper. On the other hand, if the fundamental
eigenspaces of $M[x(\cdot)]$ for different paths maximizing
$\mu_{1}[x(\cdot)]$ have a non trivial intersection, then for all the
realizations with $\bm{\mathfrak{s}}$ in the intersection, the instanton has
more than one ridge path. This case corresponds to multi-filament instantons.

##  Appendix B Equivalence of the Fourier and convolution representations of
$\bm{S_{\rm inst}}$

In this appendix, we prove the equivalence of the expressions of $S_{\rm
inst}^{x_{\rm inst}(\cdot)}(x,z)$ in Eqs. (49) and (51). Permuting the sum and
the integral on the right-hand side of Eq. (51) and using the Fourier
decomposition (41) for $C(x-x_{\rm inst}(z^{\prime}),z,z^{\prime})$, one
readily finds that the equation (51) can be rewritten as

$S_{\rm inst}^{x_{\rm inst}(\cdot)}(x,z)=c_{1}\Omega_{1}(x,z),$ (73)

with

$\Omega_{1}(x,z)=\sum_{(n,j)\in\mathcal{I}}\mathfrak{e}^{(1)}_{(n,j)}\sqrt{\frac{\sigma_{(n,j)}}{\ell\,\mu_{\rm
max}}}\,{\rm e}^{2i\pi nx/\ell}\Phi_{(n,j)}(z),$ (74)

where $\bm{\mathfrak{e}}^{(1)}$ is a vector defined by its components

$\mathfrak{e}^{(1)}_{(n,j)}=\sqrt{\frac{\sigma_{(n,j)}}{\ell\,\mu_{\rm
max}}}\,\int_{0}^{L}{\rm e}^{-2i\pi nx_{\rm
inst}(z^{\prime})/\ell}\Phi_{(n,j)}(z^{\prime})^{\ast}\phi_{1}(z^{\prime})\,dz^{\prime}.$
(75)

Showing that the equation (49) can also be written in the form of Eq. (73)
requires a little more work. From Eqs. (41), (42), and (75) it can be checked
that

$\displaystyle\left(M[x_{\rm
inst}(\cdot)]\bm{\mathfrak{e}}^{(1)}\right)_{(n,j)}=\sum_{(m,k)\in\mathcal{I}}M_{(n,j)(m,k)}[x_{\rm
inst}(\cdot)]\,\mathfrak{e}^{(1)}_{(m,k)}$
$\displaystyle=\sqrt{\frac{\sigma_{(n,j)}}{\ell\,\mu_{\rm
max}}}\,\int_{0}^{L}{\rm e}^{-2i\pi nx_{\rm
inst}(z)/\ell}\Phi_{(n,j)}(z)^{\ast}\,\langle z|T_{x_{\rm
inst}(\cdot)}|\phi_{1}\rangle\,dz$ $\displaystyle=\mu_{\rm
max}\,\mathfrak{e}^{(1)}_{(n,j)},$ (76)

and

$\displaystyle\|\bm{\mathfrak{e}}^{(1)}\|^{2}=\sum_{(n,j)\in\mathcal{I}}|\mathfrak{e}^{(1)}_{(n,j)}|^{2}=\frac{1}{\mu_{\rm
max}}\,\langle\phi_{1}|T_{x_{\rm inst}(\cdot)}|\phi_{1}\rangle$ (77)
$\displaystyle=\langle\phi_{1}|\phi_{1}\rangle=1,$

which means that $\bm{\mathfrak{e}}^{(1)}$ is the normalized fundamental
eigenvector of $M[x_{\rm inst}(\cdot)]$. Writing

$\mathfrak{s}_{(n,j)}=\frac{1}{\sqrt{\mu_{\rm
max}}}\,c_{1}\mathfrak{e}^{(1)}_{(n,j)}\ {\rm with}\ c_{1}=\sqrt{\mu_{\rm
max}}\,\sum_{(n,j)\in\mathcal{I}}\mathfrak{s}_{(n,j)}\mathfrak{e}^{(1)\,\ast}_{(n,j)},$
(78)

on the right-hand side of Eq. (49), one obtains the same equation (73), as
expected, which proves the equivalence of Eqs. (49) and (51) with
$\mathfrak{s}_{(n,j)}$ and $c_{1}$ related to each other by Eq. (78).

##  Appendix C Limit of $\bm{\eta^{-1}\ln A(\eta)}$ and
$\bm{\partial_{\eta}\ln A(\eta)}$ as $\bm{\eta\to+\infty}$

In this appendix we derive the two limits in Eq. (59). We will use the
convolution representation (51). To make the dependence of $S_{\rm
inst}^{x_{\rm inst}(\cdot)}$ on $c_{1}$ explicit we write $S_{\rm
inst}^{x_{\rm inst}(\cdot)}(x,z)\equiv S_{\rm inst}^{x_{\rm
inst}(\cdot)}(x,z,c_{1})=\sqrt{\eta}\,{\rm e}^{i\arg(c_{1})}\,S_{\rm
inst}^{x_{\rm inst}(\cdot)}(x,z,1)$, where $\eta=|c_{1}|^{2}$. Deriving the
Feynman-Kac path-integral representation of $\varphi_{\rm inst}^{x_{\rm
inst}(\cdot)}(0,L)$,

$\varphi_{\rm inst}^{x_{\rm inst}(\cdot)}(0,L)=\int_{x(\cdot)\in B(0,L)}{\rm
e}^{\int_{0}^{L}\left[\frac{im}{2}\dot{x}(\tau)^{2}+g\eta|S_{\rm inst}^{x_{\rm
inst}(\cdot)}(x(\tau),\tau,1)|^{2}\right]\,d\tau}\mathscr{D}x,$ (79)

with respect to $\eta$ and using the fact that $\int_{0}^{L}|S_{\rm
inst}^{x_{\rm inst}(\cdot)}(x_{\rm inst}(\tau),\tau,1)|^{2}d\tau=1$, one gets

$\displaystyle\partial_{\eta}\varphi_{\rm inst}^{x_{\rm inst}(\cdot)}(0,L)$
$\displaystyle=$ $\displaystyle g\int_{x(\cdot)\in
B(0,L)}\left(\int_{0}^{L}|S_{\rm inst}^{x_{\rm
inst}(\cdot)}(x(\tau),\tau,1)|^{2}d\tau\right)$ (80)
$\displaystyle\times\,{\rm
e}^{\int_{0}^{L}\left[\frac{im}{2}\dot{x}(\tau)^{2}+g\eta|S_{\rm inst}^{x_{\rm
inst}(\cdot)}(x(\tau),\tau,1)|^{2}\right]\,d\tau}\mathscr{D}x$
$\displaystyle\sim$ $\displaystyle g\int_{x(\cdot)\in
B(0,L)}\left(\int_{0}^{L}|S_{\rm inst}^{x_{\rm inst}(\cdot)}(x_{\rm
inst}(\tau),\tau,1)|^{2}d\tau\right)$ $\displaystyle\times\,{\rm
e}^{\int_{0}^{L}\left[\frac{im}{2}\dot{x}(\tau)^{2}+g\eta|S_{\rm inst}^{x_{\rm
inst}(\cdot)}(x(\tau),\tau,1)|^{2}\right]\,d\tau}\mathscr{D}x$
$\displaystyle=$ $\displaystyle g\int_{x(\cdot)\in B(0,L)}{\rm
e}^{\int_{0}^{L}\left[\frac{im}{2}\dot{x}(\tau)^{2}+g\eta|S_{\rm inst}^{x_{\rm
inst}(\cdot)}(x(\tau),\tau,1)|^{2}\right]\,d\tau}\mathscr{D}x$
$\displaystyle=$ $\displaystyle g\varphi_{\rm inst}^{x_{\rm
inst}(\cdot)}(0,L)\ \ \ \ \ (\eta\to+\infty),$

from which it follows that

$\partial_{\eta}\ln|\varphi_{\rm inst}^{x_{\rm inst}(\cdot)}(0,L)|^{2}=2{\rm
Re}\left(\frac{\partial_{\eta}\varphi_{\rm inst}^{x_{\rm
inst}(\cdot)}(0,L)}{\varphi_{\rm inst}^{x_{\rm inst}(\cdot)}(0,L)}\right)\sim
2g\ \ \ \ \ (\eta\to+\infty).$ (81)

Thus, for all $\varepsilon>0$ there is $\eta_{0}>0$ such that for every
$\eta\geq\eta_{0}$,

$2g(1-\varepsilon)\leq\partial_{\eta}\ln|\varphi_{\rm inst}^{x_{\rm
inst}(\cdot)}(0,L)|^{2}\leq 2g(1+\varepsilon).$ (82)

Writing $|\varphi_{\rm inst}^{x_{\rm inst}(\cdot)}(0,L)|^{2}=A(\eta)\,{\rm
e}^{2g\eta}$ in Eq. (82), one obtains

$-2g\varepsilon\leq\partial_{\eta}\ln A(\eta)\leq 2g\varepsilon,$ (83)

for every $\eta\geq\eta_{0}$, and since $\varepsilon$ can be taken arbitrarily
small, Eq. (83) reduces to

$\lim_{\eta\to+\infty}\partial_{\eta}\ln A(\eta)=0,$ (84)

which is the second limit in Eq. (59). To get the first limit, we integrate
Eq. (83) from $\eta_{0}$ to any $\eta>\eta_{0}$, which yields

$-2g\varepsilon\eta+K_{+}(\varepsilon)\leq\ln A(\eta)\leq
2g\varepsilon\eta+K_{-}(\varepsilon),$ (85)

with $K_{\pm}(\varepsilon)=\ln A(\eta_{0})\pm 2g\varepsilon\eta_{0}$. Note
that $\ln A(\eta_{0})$ exists, otherwise $\ln A(\eta)$ would have a vertical
asymptote at $\eta=\eta_{0}$, in contradiction with Eq. (83). It remains to
divide Eq. (85) by $\eta$:

$-2g\varepsilon-\frac{|K_{+}(\varepsilon)|}{\eta}\leq\frac{1}{\eta}\ln
A(\eta)\leq 2g\varepsilon+\frac{|K_{-}(\varepsilon)|}{\eta},$ (86)

where we have used $K_{+}(\varepsilon)\geq-|K_{+}(\varepsilon)|$ and
$K_{-}(\varepsilon)\leq|K_{-}(\varepsilon)|$. Now, for $\eta$ large enough,
namely
$\eta\geq\max[\eta_{0},\varepsilon^{-1}|K_{+}(\varepsilon)|,\varepsilon^{-1}|K_{-}(\varepsilon)|]$,
Eq. (86) gives

$-3g\varepsilon\leq\frac{1}{\eta}\ln A(\eta)\leq 3g\varepsilon,$ (87)

and since $\varepsilon$ can be taken arbitrarily small, one finally obtains

$\lim_{\eta\to+\infty}\frac{1}{\eta}\ln A(\eta)=0,$ (88)

which is the first limit in Eq. (59).

We now prove the relation $\ln A((2g)^{-1}\ln U)^{1/2\mu_{\rm max}g}=o(\ln U)$
used in Eq. (60). For all $\varepsilon>0$ there is $\eta_{0}>0$ such that for
every $\eta\geq\eta_{0}$,

$\exp(-\varepsilon\eta)\leq A(\eta)\leq\exp(\varepsilon\eta),$ (89)

whence,

$\exp\left(-\frac{\varepsilon\eta}{2\mu_{\rm max}g}\right)\leq
A(\eta)^{1/2\mu_{\rm max}g}\leq\exp\left(\frac{\varepsilon\eta}{2\mu_{\rm
max}g}\right),$ (90)

and, taking the logarithm,

$-\frac{\varepsilon\eta}{2\mu_{\rm max}g}\leq\ln A(\eta)^{1/2\mu_{\rm
max}g}\leq\frac{\varepsilon\eta}{2\mu_{\rm max}g},$ (91)

for every $\eta\geq\eta_{0}$. Since $\varepsilon$ can be taken arbitrarily
small, Eq. (91) reduces to

$\lim_{\eta\to+\infty}\frac{1}{\eta}\ln A(\eta)^{1/2\mu_{\rm max}g}=0,$ (92)

which means that $\ln A(\eta)^{1/2\mu_{\rm max}g}=o(\eta)$, as announced.

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