# Scaling properties of (2+1) directed polymers in the low temperature limit

Victor Dotsenko  LPTMC, Sorbonne Université, Paris, France

###### Abstract

In terms of the replica method we consider the low temperature limit of (2+1)
directed polymers in a random potential. The proposed approach allows to
compute the scaling exponent $\theta$ of the free energy fluctuations as well
as the left tail of its probability distribution function. It is argued that
$\theta=1/4$ which is slightly different from the zero-temperature numerical
value which is close to $0.241$.

###### pacs:

05.20.-y 75.10.Nr 74.25.Qt 61.41.+e

##  I Introduction

In contrast to the achievements in the studies of one-dimensional directed
polymers as well as the other systems belonging to the so called KPZ
universality class KPZ ; Corwin ; Borodin ; Rev ; Takeuchi , not so much is
achieved in understanding of the scaling properties of the so called (2+1)
model of directed polymers which describes the fluctuations of an elastic
string directed along the time axes and which passes through a random medium
in the three-dimensional space (2+1)-2009 ; (2+1)-2017a ; (2+1)-2017b ;
(2+1)-2019 ; (2+1)-2020 ; (2+1)-exper-2015 . Extensive numerical simulations
as well as experimental studies rather convincingly demonstrate that in the
limit of large times $t$ the free energy fluctuations of such polymers scale
as $t^{\theta}$ with the scaling exponent $\theta\simeq 0.241$ which is very
close but not equal to $1/4$ numerics1 ; numerics2 ; numerics3 . On the other
hand, in the recent analytical study (2+1)-highT  it has been argued that in
the high-temperature limit this scaling exponent is equal to $\theta=0.5$. If
this statement is correct, it would mean that the scaling exponent of the
(2+1) system is not universal being temperature dependent in the drastic
difference with the corresponding (1+1) model where this exponent is universal
and equal to $1/3$ at all temperatures.

In this paper in terms of the replica technique we study the scaling
properties of the two-dimensional directed polymers in the low-temperature
limit. The idea is to estimate the replica partition function $Z(N,t)$ (which
is the disorder average of the $N$-th power of the partition function) in the
limit of $N\gg 1$ which makes possible to reconstruct the left tail asymptotic
of the free energy distribution function $P(F\to-\infty)$. Assuming that this
(unknown) distribution function is defined by the only energy scale we find
that the time scaling exponent of the free energy fluctuations is
$\theta=1/4$. We see that this value is somewhat different from the zero
temperature numerical results. To explain this discrepancy I would suggest
that in the numerical simulations the exact value of the scaling exponent
$\theta=1/4$ is distorted by the presence of a logarithmic prefactor which is
common situation for the two-dimensional systems.

In Section II we define the model and describe the general ideas of the
considered replica approach. In Section III we present the main lines of the
”energy optimization method” used for estimation of the replica partition
function in the large $N$ limit. In Section IV we demonstrate how this method
works in the case of (1+1) directed polymers and show that here it perfectly
reproduces well known exact results. In section V we concentrate on (2+1)
directed polymer model for which up to now no exact results are known. Here we
demonstrate that in the low temperature limit and at $N\gg 1$ the replica
partition function $Z(N,t)\sim\exp\bigl{\\{}u\,(\beta\,\epsilon)^{-2}\,(\beta
N)^{4}\,t\bigr{\\}}$, where $\beta$ is the inverse temperature, $u$ is the
strength of the random potential and $\epsilon$ is its spatial correlation
length (see eqs.(3)-(4)). Correspondingly, for the left tail of the free
energy distribution function one obtains

$P(F\to-\infty)\;\sim\;\exp\biggl{\\{}-\biggl{(}\frac{|F|}{\bigl{(}\sqrt{u}/\beta\epsilon\bigr{)}^{1/2}\;t^{1/4}}\biggr{)}^{4/3}\biggr{\\}}$
(1)

so that the free energy fluctuations scales as $|F|\,\sim\,t^{1/4}$. Finally,
Section VI is devoted to the discussion of the obtained results.

##  II The model and the replicas approach

We consider the model of directed polymers defined in terms of an elastic
string described by the two-dimensional vector
$\boldsymbol{\phi}(\tau)\equiv\bigl{(}\phi_{x}(\tau),\,\phi_{y}(\tau)\bigr{)}$
directed along the $\tau$-axes within an interval $[0,t]$ which passes through
a random medium described by a random potential $V(\boldsymbol{\phi},\tau)$.
The energy of a given polymer’s trajectory $\boldsymbol{\phi}(\tau)$ is

$H[\boldsymbol{\phi}(\tau);V]=\int_{0}^{t}d\tau\biggl{[}\frac{1}{2}\Bigl{(}\partial_{\tau}\boldsymbol{\phi}(\tau)\Bigr{)}^{2}+V\bigl{(}\boldsymbol{\phi}(\tau),\tau\bigr{)}\biggr{]};$
(2)

Here the disorder potential $V[\boldsymbol{\phi},\tau]$ is supposed to be
Gaussian distributed with a zero mean $\overline{V(\boldsymbol{\phi},\tau)}=0$
and the correlation function

$\overline{V(\boldsymbol{\phi},\tau)V(\boldsymbol{\phi}^{\prime},\tau^{\prime})}=u\,\delta(\tau-\tau^{\prime})\delta^{(2d)}_{\epsilon}(\boldsymbol{\phi}-\boldsymbol{\phi}^{\prime})$
(3)

The parameter $u$ is the strength of the disorder and
$\delta^{(2d)}_{\epsilon}(\boldsymbol{\phi})$ is the two-dimensional ”finite-
size $\delta$-function”:

$\delta^{(2d)}_{\epsilon}(\boldsymbol{\phi})\;=\;\left\\{\begin{array}[]{ll}\frac{1}{\pi\epsilon^{2}}\;,\;\;\;\;\mbox{for}\;|\boldsymbol{\phi}|\,\leq\,\epsilon\\\
\\\
0\;,\;\;\;\;\;\;\;\mbox{for}\;|\boldsymbol{\phi}|\,>\,\epsilon\end{array}\right.$
(4)

We do not introduce here ”true” (zero-size) two-dimensional $\delta$-function
for the reasons which will be explained later.

For a given realization of the random potential $V(\boldsymbol{\phi},\tau)$
the partition function of the considered system (with fixed boundary
conditions) is

$Z({\bf r},t)=\int_{\boldsymbol{\phi}(0)=\bf{0}}^{\boldsymbol{\phi}(t)={\bf
r}}{\cal D}\boldsymbol{\phi}(\tau)\exp\bigl{\\{}-\beta
H[\boldsymbol{\phi}(\tau),V]\bigr{\\}}\;=\;\exp\bigl{\\{}-\beta F({\bf
r},t)\bigr{\\}}$ (5)

where $\beta$ is the inverse temperature, $F({\bf r},t)$ is the free energy
which is a random quantity and the integration is taken over all trajectories
$\boldsymbol{\phi}(\tau)$ starting at $\bf{0}$ (at $\tau=0$) and ending at a
point ${\bf r}$ (at $\tau=t$).

For simplicity, in what follows we are going to consider the problem with the
zero boundary conditions:
$\boldsymbol{\phi}(\tau=0)=\boldsymbol{\phi}(\tau=t)={\bf 0}$. The free energy
probability distribution function $P(F)$ of this system can be studied in
terms of the integer moments of the above partition function, eq.(5):

$\overline{Z^{N}}\equiv
Z(N,t)\;=\;\prod_{a=1}^{N}\int_{\boldsymbol{\phi}_{a}(0)=0}^{\boldsymbol{\phi}_{a}(t)=0}{\cal
D}\boldsymbol{\phi}_{a}(\tau)\;\overline{\Biggl{(}\exp\Bigl{\\{}-\beta\sum_{a=1}^{N}H[\boldsymbol{\phi}_{a}(\tau),V]\Bigr{\\}}\Biggr{)}}\;=\;\int_{-\infty}^{+\infty}dF\,P(F)\,\exp\bigl{\\{}-\beta
NF\bigr{\\}}$ (6)

where $\overline{(...)}$ denotes the averaging over the random potentials
$V[\boldsymbol{\phi},\tau]$ Performing this simple Gaussian averaging we get

$Z(N,t)\;=\;\prod_{a=1}^{N}\int_{\boldsymbol{\phi}_{a}(0)=0}^{\boldsymbol{\phi}_{a}(t)=0}{\cal
D}\boldsymbol{\phi}_{a}(\tau)\;\exp\Bigl{\\{}-\beta
H_{N}[\boldsymbol{\phi}_{1}(\tau),\,\boldsymbol{\phi}_{2}(\tau),\,...\,,\boldsymbol{\phi}_{N}(\tau)]\Bigr{\\}}$
(7)

where

$\beta
H_{N}[\boldsymbol{\phi}_{1}(\tau),\,\boldsymbol{\phi}_{2}(\tau),\,...\,,\boldsymbol{\phi}_{N}(\tau)]=\int_{0}^{t}d\tau\Biggl{[}\frac{1}{2}\beta\sum_{a=1}^{N}\Bigl{(}\partial_{\tau}\boldsymbol{\phi}_{a}(\tau)\Bigr{)}^{2}\;-\;\frac{1}{2}\beta^{2}\,u\,\sum_{a,b=1}^{N}\delta^{(2d)}_{\epsilon}\bigl{(}\boldsymbol{\phi}_{a}(\tau)-\boldsymbol{\phi}_{b}(\tau)\bigr{)}\Biggr{]};$
(8)

is the replica Hamiltonian which describes $N$ elastic strings
$\bigl{\\{}\boldsymbol{\phi}_{1}(\tau),\,\boldsymbol{\phi}_{2}(\tau),\,...\,,\boldsymbol{\phi}_{N}(\tau)\bigr{\\}}$
with the attractive interactions
$\delta^{(2d)}_{\epsilon}\bigl{(}\boldsymbol{\phi}_{a}-\boldsymbol{\phi}_{b}\bigr{)}$,
eq.(4). To compute the replica partition function $Z(N,t)$, eqs.(7)-(8), one
introduces the function:

$\Psi({\bf r}_{1},\,{\bf r}_{2},\,...\,{\bf
r}_{N};\;t)\;=\;\prod_{a=1}^{N}\int_{\boldsymbol{\phi}_{a}(0)=\bf{0}}^{\boldsymbol{\phi}_{a}(t)={\bf
r}_{a}}{\cal D}\boldsymbol{\phi}_{a}(\tau)\;\exp\Bigl{\\{}-\beta
H_{N}[\boldsymbol{\phi}_{1}(\tau),\,\boldsymbol{\phi}_{2}(\tau),\,...\,,\boldsymbol{\phi}_{N}(\tau)]\Bigr{\\}}$
(9)

such that

$Z(N,t)\,=\,\Psi({\bf r}_{1},\,{\bf r}_{2},\,...\,{\bf
r}_{N};\,t)\big{|}_{{\bf r}_{a}=0}$ (10)

One can easily show that $\Psi({\bf r}_{1},\,{\bf r}_{2},\,...\,{\bf
r}_{N};\,t)$ is the wave function of N quantum bosons defined by the imaginary
time Schrödinger equation

$\beta\frac{\partial}{\partial t}\Psi({\bf r}_{1},\,{\bf r}_{2},\,...\,{\bf
r}_{N};\;t)\;=\;\frac{1}{2}\sum_{a=1}^{N}\,\Delta_{a}\Psi({\bf r}_{1},\,{\bf
r}_{2},\,...\,{\bf
r}_{N};\;t)\;+\;\frac{1}{2}\,\beta^{3}u\,\sum_{a,b=1}^{N}\delta^{(2d)}_{\epsilon}({\bf
r}_{a}-{\bf r}_{b})\,\Psi({\bf r}_{1},\,{\bf r}_{2},\,...\,{\bf r}_{N};\;t)$
(11)

where $\Delta_{a}$ is the two-dimensional Laplacian with respect to the
coordinate ${\bf r}_{a}$. The corresponding eigenvalue equation for the
eigenfunctions $\psi({\bf r}_{1},\,{\bf r}_{2},\,...\,{\bf r}_{N})$, defined
by the relation

$\Psi({\bf r}_{1},\,{\bf r}_{2},\,...\,{\bf r}_{N};\;t)\;=\;\psi({\bf
r}_{1},\,{\bf r}_{2},\,...\,{\bf r}_{N})\,\exp\bigl{\\{}-t\,E_{N}\bigr{\\}}$
(12)

reads:

$-2\beta\,E_{N}\,\psi({\bf r}_{1},\,{\bf r}_{2},\,...\,{\bf
r}_{N})\;=\;\sum_{a=1}^{N}\,\Delta_{a}\psi({\bf r}_{1},\,{\bf
r}_{2},\,...\,{\bf
r}_{N})\;+\;\kappa\,\sum_{a,b=1}^{N}\delta^{(2d)}_{\epsilon}({\bf r}_{a}-{\bf
r}_{b})\,\psi({\bf r}_{1},\,{\bf r}_{2},\,...\,{\bf r}_{N})$ (13)

where $\kappa=\beta^{3}u$.

It is at this stage that we are facing the crucial difference of the
considered problem with the corresponding (1+1) one (see Section III). The
general solution of the one-dimensional counterpart of eq.(13) is given by the
Bethe ansatz wave function Lieb-Liniger ; McGuire ; Yang  which is valid only
for $U(x)\,=\,\delta(x)$ and which is based on the exact two-particle wave
functions ($N=2$) solution exhibiting finite value energy $E_{N=2}$. It is
this fundamental property of (1+1) problem which eventually allows to derive
the Tracy-Widom distribution for the free energy fluctuation KPZ-TW1a ; KPZ-
TW1b ; KPZ-TW1c ; KPZ-TW2 ; LeDoussal-2010 ; VD-TW-2010-1 ; VD-TW-2010-2 . The
situation in the (2+1) case, eq.(13), is much more complicated. First of all,
in two dimensions there exists no finite two-particle solution for the true
$\delta$-function potential $\lim_{\epsilon\to 0}\delta^{(2d)}_{\epsilon}({\bf
r})$, eq(4). Indeed, for $N=2$ eq.(13) (where for simplicity we take $\beta=1$
and $u=\pi$) reduces to

$\psi^{\prime\prime}(r)\,+\,\frac{1}{r}\psi^{\prime}(r)\,+\,\bigl{(}\pi\delta^{(2d)}_{\epsilon}({\bf
r})\,+\,2E_{2}\bigr{)}\,\psi(r)\;=\;0$ (14)

where $r=|{\bf r}_{1}-{\bf r}_{2}|$. It can be easily shown that at
$\epsilon\ll 1$ the ground state solution of this equation (with $E_{2}<0$) is

$\psi(r)\;\simeq\;\left\\{\begin{array}[]{ll}\psi_{0}\,J_{0}\bigl{(}\sqrt{1-2\epsilon^{2}|E_{2}|}\;r/\epsilon\bigr{)}\;,\;\;\;\;\mbox{for}\;r\,\leq\,\epsilon\\\
\\\
\psi_{1}\,K_{0}\bigl{(}\sqrt{2\epsilon^{2}|E_{2}|}\;r/\epsilon\bigr{)}\;,\;\;\;\;\;\;\;\;\;\mbox{for}\;r\,>\,\epsilon\end{array}\right.$
(15)

where $J_{0}(x)$ and $K_{0}(x)$ are Bessel and Macdonald functions
correspondingly, and (irrelevant) constants $\psi_{0}$ and $\psi_{1}$ are
related by the condition
$\psi_{0}\,J_{0}\bigl{(}\sqrt{1-2\epsilon^{2}|E_{2}|}\bigr{)}\;=\;\psi_{1}\,K_{0}\bigl{(}\sqrt{2\epsilon^{2}|E_{2}|}\bigr{)}$.
The ground state energy of this two-particle state is

$E_{2}(\epsilon)\;\simeq\;-\frac{1}{2\epsilon^{2}}\,x_{*}$ (16)

where the number $x_{*}\simeq 0.04$ is the solution of the equation

$\frac{J_{0}^{\prime}\bigl{(}\sqrt{1-x}\bigr{)}\;\sqrt{1-x}}{J_{0}\bigl{(}\sqrt{1-x}\bigr{)}}\;=\;\frac{K_{0}^{\prime}\bigl{(}\sqrt{x}\bigr{)}\;\sqrt{x}}{K_{0}\bigl{(}\sqrt{x}\bigr{)}}$
(17)

We see that in the limit $\epsilon\to 0$ (where $\delta^{(2d)}_{\epsilon}({\bf
r})$ turns into the ”true” $\delta$-function) the ground state energy of this
state $E_{2}(\epsilon\to 0)\,\to\,-\infty$. This simple exercise clearly
demonstrate that first, consideration of the $N$-particle problem, eq.(13)
with the true (zero-size) two-dimensional $\delta$-function makes no sense and
second, the ground state energy of this $N$-particle problem must explicitly
depend on the (non-zero) size $\epsilon$ of the interaction potential
$\delta^{(2d)}_{\epsilon}({\bf r})$, eq.(4).

###  II.1 Free energy probability distribution function in the replica
approach

Free energy probability distribution function (PDF) $P(F)$ and the replica
partition function are related according to eq.(6):

$\int_{-\infty}^{+\infty}dF\,P(F)\,\exp\bigl{\\{}-\beta
NF\bigr{\\}}\;=\;Z(N,t)$ (18)

The problem is that to recover $P(F)$ from the above equation (e.g. by inverse
Laplace transform) one has to derive the exact solution for the replica
partition function $Z(N,t)$ which is hardly possible for the model under
consideration. However if we do not pretend to obtain an entire PDF $P(F)$,
but we are interested to know only its left tail, $P(F\to-\infty)$, then the
situation somewhat simplifies. The point is that the asymptotics
$P(F\to-\infty)$ is defined by the behavior of the replica partition function
$Z(N,t)$ in the limit $N\to\infty$ only.

For example, let us suppose that in the limit $N\to\infty$ the replica
partition function, eqs.(10), (12) has the form

$Z(N,t)\;\sim\;\exp\Bigl{\\{}g\,(\beta N)^{k}\,t\Bigr{\\}}$ (19)

where the factor $g$ depends on the parameters of the model. In this case it
can be easily shown that the left tail of the function $P(F)$ has stretched-
exponential form,

$P(F\to-\infty)\;\sim\;\exp\Bigl{\\{}-\gamma\,|F|^{\alpha}\Bigr{\\}}$ (20)

where it is supposed that $\alpha>1$. Indeed, substituting eqs.(19) and (20)
into eq.(18) we get

$\int_{-\infty}^{0}dF\,P(F)\,\exp\bigl{\\{}\beta
N|F|\bigr{\\}}\;+\;\int_{0}^{+\infty}dF\,P(F)\,\exp\bigl{\\{}-\beta
NF\bigr{\\}}\;\sim\;\exp\Bigl{\\{}g\,(\beta N)^{k}\,t\Bigr{\\}}$ (21)

It is clear that due to the factor $\exp\bigl{\\{}-\beta NF\bigr{\\}}$ the
second integral in the l.h.s of the above relation vanishes in the limit
$N\to+\infty$, while the first integral (over negative values of $F$) is
dominated by the saddle-point contribution of left tail of the function
$P(F)$, namely

$\int_{-\infty}^{0}dF\,\exp\bigl{\\{}-\gamma\,|F|^{\alpha}\,+\,\beta
N|F|\bigr{\\}}\;\sim\;\exp\Bigl{\\{}g\,(\beta N)^{k}\,t\Bigr{\\}}$ (22)

Formally the asymptotic form, eq.(20), is valid only for large (negative)
values of $F$ so that the integration here can not be extended up to zero.
Nevertheless, since the saddle-point value of the free energy,

$|F_{*}|\;=\;\biggl{(}\frac{\beta
N}{\alpha\,\gamma}\biggr{)}^{\frac{1}{\alpha-1}}$ (23)

is big in the limit $N\gg 1$, the contribution of the integration over finite
(negative) values of $F$ (where PDF $P(F)$ is not known) can be neglected
compared to the saddle-point contribution in the asymptotic region
$F\to-\infty$. Thus, substituting this saddle-point value into the exponential
in the l.h.s of eq.(22) we get

$\exp\Bigl{\\{}\frac{\alpha}{\alpha-1}\bigl{(}\alpha\gamma\bigr{)}^{-\frac{1}{\alpha-1}}\,\bigl{(}\beta
N\bigr{)}^{\frac{\alpha}{\alpha-1}}\Bigr{\\}}\;\sim\;\exp\Bigl{\\{}g\,(\beta
N)^{k}\,t\Bigr{\\}}$ (24)

From this relation we immediately obtain

$\alpha\;=\;\frac{k}{k-1}$ (25)

and

$\gamma\;=\;\frac{k-1}{k}\,\bigl{(}k\,g\,t\bigr{)}^{-\frac{1}{k-1}}$ (26)

Substituting these values of $\alpha$ and $\gamma$ into eq.(20) we get the
following left asymptotics of the free energy probability distribution
function

$P(F\to-\infty)\;\sim\;\exp\biggl{\\{}-\frac{k-1}{k}\biggl{(}\frac{|F|}{\bigl{(}k\,g\,t\bigr{)}^{1/k}}\biggr{)}^{\frac{k}{k-1}}\biggr{\\}}$
(27)

Now, if we admit that the free energy fluctuations of the considered system
are described by a universal probability distribution function, then we can
conclude that the time scaling of these fluctuations can be represented as

$F\;\sim\;f\cdot t^{1/k}$ (28)

where the random quantity $f\sim 1$ is described by a probability distribution
function ${\cal P}(f)$ whose left tail is

${\cal
P}(f\to-\infty)\;\sim\;\exp\Bigl{\\{}-\mbox{(const)}\,|f|^{k/(k-1)}\Bigr{\\}}$
(29)

while its entire form remains unknown.

##  III Replica energy balance method

In view of the absence of the exact solution for the $N$-particle two-
dimensional wave function $\psi({\bf r}_{1},\,...\,{\bf r}_{N})$ we propose
the following ”heuristic” strategy of calculations of the replica energy
$E_{N}$ defined by eq.(13) in the limit $N\gg 1$. Let us multiply eq.(13) by
$\psi({\bf r}_{1},\,...\,{\bf r}_{N})\;\delta\bigl{(}\sum_{a=1}^{N}{\bf
r}_{a}\bigr{)}$ and integrate over all particle’s positions:

$\displaystyle-2\beta\,E_{N}\,\int\,d^{2}r_{1}...d^{2}r_{N}\delta\Bigl{(}\sum_{a=1}^{N}{\bf
r}_{a}\Bigr{)}\psi^{2}({\bf r}_{1},\,...\,{\bf
r}_{N})=\sum_{a=1}^{N}\,\int\,d^{2}r_{1}...d^{2}r_{N}\;\delta\Bigl{(}\sum_{a=1}^{N}{\bf
r}_{a}\Bigr{)}\psi({\bf r}_{1},\,...\,{\bf r}_{N})\Delta_{a}\psi({\bf
r}_{1},\,...\,{\bf r}_{N})$
$\displaystyle+\kappa\,\sum_{a\not=b}^{N}\int\,d^{2}r_{1}...d^{2}r_{N}\delta\Bigl{(}\sum_{a=1}^{N}{\bf
r}_{a}\Bigr{)}\delta^{(2d)}_{\epsilon}({\bf r}_{a}-{\bf r}_{b})\,\psi^{2}({\bf
r}_{1},\,...\,{\bf
r}_{N})+\frac{\kappa}{\pi\epsilon^{2}}\,N\int\,d^{2}r_{1}...d^{2}r_{N}\delta\Bigl{(}\sum_{a=1}^{N}{\bf
r}_{a}\Bigr{)}\psi^{2}({\bf r}_{1},\,...\,{\bf r}_{N})$ (30)

Taking into account the symmetry of the wave function with respect to the
particles permutations the above equation can be rewritten as follows

$\displaystyle
2\beta\,\tilde{E}_{N}\,\int\,d^{2}r_{1}...d^{2}r_{N}\;\delta\Bigl{(}\sum_{a=1}^{N}{\bf
r}_{a}\Bigr{)}\psi^{2}({\bf r}_{1},\,...\,{\bf
r}_{N})=N\,\int\,d^{2}r_{1}...d^{2}r_{N}\;\delta\Bigl{(}\sum_{a=1}^{N}{\bf
r}_{a}\Bigr{)}\Bigl{(}\nabla_{1}\psi({\bf r}_{1},\,...\,{\bf
r}_{N})\Bigr{)}^{2}$
$\displaystyle-\;\kappa\,N(N-1)\int\,d^{2}r_{1}...d^{2}r_{N}\;\delta\Bigl{(}\sum_{a=1}^{N}{\bf
r}_{a}\Bigr{)}\delta^{(2d)}_{\epsilon}({\bf r}_{1}-{\bf r}_{2})\,\psi^{2}({\bf
r}_{1},\,...\,{\bf r}_{N})$ (31)

where

$\tilde{E}_{N}\;=\;E_{N}\;+\;\frac{\kappa N}{2\beta\,\epsilon^{2}}$ (32)

Let us suppose that the wave function of the present system is characterized
by the only spatial scale $R$ (which will be defined later by minimizing the
energy $\tilde{E}_{N}$), $\psi({\bf r}_{1},\,...\,{\bf r}_{N})\;\to\;\psi({\bf
r}_{1}/R,\,...\,{\bf r}_{N}/R)$. Taking into account the properties of wave
function of the two-particle system, eqs.(15), where the Macdonald function

$K_{0}(x)\;\sim\;\left\\{\begin{array}[]{ll}\ln(1/x)\;,\;\;\;\;\;\;\;\mbox{at}\;x\ll
1\\\ \\\ \exp\\{-x\\}\;,\;\;\;\;\mbox{at}\;x\gtrsim 1\end{array}\right.$ (33)

we will suppose that the $N$-particle wave function $\psi({\bf
r}_{1}/R,\,...\,{\bf r}_{N}/R)$ exhibits two principal properties:

(1) for any two particles $a$ and $b$, such that $\epsilon<|{\bf r}_{a}-{\bf
r}_{b}|\ll R$,

$\psi\bigl{(}{\bf r}_{1}/R,\,...,\,{\bf r}_{a}/R,\,...,\,{\bf
r}_{b}/R,\,...,\,{\bf r}_{N}/R\bigr{)}\;\sim\;\ln\Bigl{(}\frac{R}{|{\bf
r}_{a}-{\bf r}_{b}|}\Bigr{)}\times\psi\bigl{(}{\bf r}_{1}/R,\,...,\,{\bf
r}_{b}/R,\,...,\,{\bf r}_{b}/R,\,...,\,{\bf r}_{N}/R\bigr{)}$ (34)

(2) at scales $|{\bf r}_{a}-{\bf r}_{b}|\gtrsim R$,

$\psi\bigl{(}{\bf r}_{1}/R,\,...,\,{\bf
r}_{N}/R\bigr{)}\;\sim\;\exp\Bigl{\\{}-\frac{1}{R}\sum_{1\leq a<b}^{N}\,|{\bf
r}_{a}-{\bf r}_{b}|\Bigr{\\}}$ (35)

After rescaling ${\bf r}_{a}\to R\,{\bf x}_{a}$ the above wave function
$\psi\bigl{(}{\bf r}_{1}/R,\,...,\,{\bf r}_{N}/R\bigr{)}\;\to\;\psi({\bf
x}_{1},\,...\,{\bf x}_{N})$, and at scales $|{\bf x}_{a}-{\bf x}_{b}|\gtrsim
1$ instead of eq.(35) we get

$\psi({\bf x}_{1},\,...\,{\bf x}_{N})\;\sim\;\exp\Bigl{\\{}-\sum_{1\leq
a<b}^{N}\,|{\bf x}_{a}-{\bf x}_{b}|\Bigr{\\}}$ (36)

Now the relation (31) takes the form

$2\beta\tilde{E}_{N}(R)\;I_{0}(N)\;=\;N\,R^{-2}\,I_{\small{\nabla}}(N,R)\;-\;N(N-1)\,\kappa\,R^{-2}\,I_{\delta}(N,R)$
(37)

where

$I_{0}(N)\;=\;\int\,d^{2}x_{1}...d^{2}x_{N}\;\delta\Bigl{(}\sum_{a=1}^{N}{\bf
x}_{a}\Bigr{)}\psi^{2}({\bf x}_{1},\,...\,{\bf x}_{N})$ (38)
$I_{\nabla}(N,R)\;=\;\int\,d^{2}x_{1}...d^{2}x_{N}\;\delta\Bigl{(}\sum_{a=1}^{N}{\bf
x}_{a}\Bigr{)}\Bigl{(}\nabla_{1}\psi({\bf x}_{1},\,...\,{\bf
x}_{N})\Bigr{)}^{2}$ (39)
$I_{\delta}(N,R)\;=\;\int\,d^{2}x_{1}...d^{2}x_{N}\;\delta\Bigl{(}\sum_{a=1}^{N}{\bf
x}_{a}\Bigr{)}\delta^{(2d)}_{\epsilon/R}({\bf x}_{1}-{\bf
x}_{2})\,\psi^{2}({\bf x}_{1},\,...\,{\bf x}_{N})$ (40)

According to eq.(34), the leading (most divergent) contribution to the
function $I_{\nabla}(N,R)$, eq.(39), comes from the integrations over ${\bf
x}_{1}$ in the vicinities of every point ${\bf x}_{a}$ ($a=2,...N$):

$I_{\nabla}(N,R)\;\sim\;\int\,d^{2}x_{2}...d^{2}x_{N}\;\sum_{a=2}^{N}\;\int_{\frac{\epsilon}{R}<|{\bf
x}_{1}-{\bf x}_{a}|\ll 1}d^{2}x_{1}\Bigl{[}\nabla_{1}\ln\Bigl{(}\frac{1}{|{\bf
x}_{1}-{\bf x}_{a}|}\Bigr{)}\Bigr{]}^{2}\delta\Bigl{(}{\bf
x_{a}}+\sum_{b=2}^{N}{\bf x}_{b}\Bigr{)}\psi^{2}\bigl{(}{\bf x}_{a},\,{\bf
x}_{2},\,...,\,{\bf x}_{a},\,...,\,{\bf x}_{N}\bigr{)}$ (41)

Using the symmetry of the wave function $\psi({\bf x}_{1},\,...\,{\bf x}_{N})$
and redefining ${\bf x}_{1}={\bf x}_{a}+{\boldsymbol{\eta}}$ we get

$I_{\nabla}(N,R)\;\sim\;(N-1)\,\int\,d^{2}x_{2}...d^{2}x_{N}\;\int_{\epsilon/R}^{1}d\eta\,\eta\,\Bigl{[}\frac{d}{d\eta}\ln\Bigl{(}\frac{1}{\eta}\Bigr{)}\Bigr{]}^{2}\delta\Bigl{(}{\bf
x_{2}}+\sum_{b=2}^{N}{\bf x}_{b}\Bigr{)}\psi^{2}\bigl{(}{\bf x}_{2},\,{\bf
x}_{2},\,{\bf x}_{3},\,...,\,{\bf x}_{N}\bigr{)}$ (42)

or

$I_{\nabla}(N,R)\;\sim\;(N-1)\,\ln\Bigl{(}\frac{R}{\epsilon}\Bigr{)}\int\,d^{2}x_{2}...d^{2}x_{N}\;\delta\Bigl{(}{\bf
x_{2}}+\sum_{b=2}^{N}{\bf x}_{b}\Bigr{)}\,\psi^{2}\bigl{(}{\bf x}_{2},\,{\bf
x}_{2},\,{\bf x}_{3},\,...,\,{\bf x}_{N}\bigr{)}$ (43)

In a similar way we can estimate the value of the function $I_{\delta}(N,R)$,
eq.(40). Taking into account that according to eq.(34)

$\lim_{|{\bf x}_{1}-{\bf x}_{2}|\to\epsilon/R}\;\psi^{2}\bigl{(}{\bf
x}_{1},\,{\bf x}_{2},\,{\bf x}_{3},\,...,\,{\bf
x}_{N}\bigr{)}\;\sim\;\ln^{2}\Bigl{(}\frac{R}{\epsilon}\Bigr{)}\;\psi^{2}\bigl{(}{\bf
x}_{2},\,{\bf x}_{2},\,{\bf x}_{3},\,...,\,{\bf x}_{N}\bigr{)}$ (44)

and that $\int d^{2}x\,\delta^{(2d)}_{\epsilon/R}({\bf x})\,=\,1$, we obtain

$I_{\delta}(N,R)\;\sim\;\ln^{2}\Bigl{(}\frac{R}{\epsilon}\Bigr{)}\int\,d^{2}x_{2}...d^{2}x_{N}\;\delta\Bigl{(}{\bf
x_{2}}+\sum_{b=2}^{N}{\bf x}_{b}\Bigr{)}\,\psi^{2}\bigl{(}{\bf x}_{2},\,{\bf
x}_{2},\,{\bf x}_{3},\,...,\,{\bf x}_{N}\bigr{)}$ (45)

Substituting eqs.(43) and (45) into eq.(37) we find

$2\beta\tilde{E}_{N}(R)\;I_{0}(N)\;\sim\;N(N-1)\,\Bigl{[}R^{-2}\,\ln\Bigl{(}\frac{R}{\epsilon}\Bigr{)}\;-\;\kappa\,R^{-2}\,\ln^{2}\Bigl{(}\frac{R}{\epsilon}\Bigr{)}\Bigr{]}\;I_{1}(N)$
(46)

or

$\tilde{E}_{N}(R)\;\sim\;\frac{N(N-1)}{2\beta}\,\Bigl{[}R^{-2}\,\ln\Bigl{(}\frac{R}{\epsilon}\Bigr{)}\;-\;\kappa\,R^{-2}\,\ln^{2}\Bigl{(}\frac{R}{\epsilon}\Bigr{)}\Bigr{]}\;\frac{I_{1}(N)}{I_{0}(N)}$
(47)

where

$I_{1}(N)\;=\;\int\,d^{2}x_{2}...d^{2}x_{N}\;\delta\Bigl{(}{\bf
x_{2}}+\sum_{b=2}^{N}{\bf x}_{b}\Bigr{)}\,\psi^{2}\bigl{(}{\bf x}_{2},\,{\bf
x}_{2},\,{\bf x}_{3},\,...,\,{\bf x}_{N}\bigr{)}$ (48)

and $I_{0}(N)$ is given in eq.(38).

Further strategy consist in two (independent) steps: we have to find optimal
parameter $R_{*}$ which minimizes $\tilde{E}_{N}(R)$, and we have to calculate
the ratio $I_{1}(N)/I_{0}(N)$ as a function of $N$ (in the limit
$N\to\infty$). In this way we will find the energy
$E_{*}(N)\equiv\tilde{E}_{N}(R_{*})$ of considered $N$-particle system as a
function of $N$ which defines the replica partition function
$Z(N,t)\sim\exp\\{-t\,E_{*}(N)\\}$. This would be sufficient (as it was shown
in Section II) to get the scaling exponent of the free energy fluctuations as
well as the form of the left tail of its probability distribution function.
But first, let us demonstrate how the proposed method works in the case of
$(1+1)$ system where we can compare obtained results with well known exact
ones.

##  IV (1+1) directed polymers

In one dimension instead of eq.(13) we have the Schrödinger equation for the
$N$-particle wave function $\psi(r_{1},\,...\,r_{N})$ which depends on $N$
one-dimensional particles coordinates
$\\{-\infty\,<\,r_{a}\,<\,+\infty\\}\;(a=1,...,N)$:

$-2\beta\,E_{N}\,\psi(r_{1},\,...\,r_{N})\;=\;\sum_{a=1}^{N}\,\frac{\partial^{2}}{\partial
r_{a}^{2}}\,\psi(r_{1},\,...\,r_{N})\;+\;\kappa\,\sum_{a,b=1}^{N}\delta_{\epsilon}(r_{a}-r_{b})\,\psi(r_{1},\,...\,r_{N})$
(49)

where, as before, $\kappa=\beta^{3}u$ and $\delta_{\epsilon}(r)$ is the
”finite-size $\delta$-function”

$\delta_{\epsilon}(r)\;=\;\left\\{\begin{array}[]{ll}\frac{1}{2\epsilon}\;,\;\;\;\;\mbox{for}\;|r|\,\leq\,\epsilon\\\
\\\ 0\;,\;\;\;\;\;\;\mbox{for}\;|r|\,>\,\epsilon\end{array}\right.$ (50)

Multiplying eq.(49) by
$\psi(r_{1},\,...\,r_{N})\;\delta\bigl{(}\sum_{a=1}^{N}r_{a}\bigr{)}$ and
integrating over all particle’s positions instead of eq.(30) we get:

$\displaystyle
2\beta\,\tilde{E}_{N}\,\int_{-\infty}^{+\infty}\,dr_{1}...dr_{N}\,\delta\Bigl{(}\sum_{a=1}^{N}\,r_{a}\Bigr{)}\,\psi^{2}(r_{1},\,...\,r_{N})=\sum_{a=1}^{N}\,\int_{-\infty}^{+\infty}\,dr_{1}...dr_{N}\,\delta\Bigl{(}\sum_{a=1}^{N}\,r_{a}\Bigr{)}\,\Bigl{(}\frac{\partial}{\partial
r_{a}}\,\psi(r_{1},\,...\,r_{N})\Bigr{)}^{2}$
$\displaystyle-\kappa\,\sum_{a\not=b}^{N}\int_{-\infty}^{+\infty}\,dr_{1}...dr_{N}\,\delta\Bigl{(}\sum_{a=1}^{N}\,r_{a}\Bigr{)}\,\delta_{\epsilon}(r_{a}-r_{b})\,\psi^{2}(r_{1},\,...\,r_{N})$
(51)

where

$\tilde{E}_{N}\;=\;E_{N}\;+\;\frac{\kappa N}{4\beta\,\epsilon}$ (52)

Supposing, as before, that the wave function of the present system is
characterized by the only spatial scale $R$ and taking into account the
properties of the two-particle solution we admit that the $N$-particle wave
function has the Bethe anzats structure,

$\psi(r_{1},\,...\,r_{N})\;=\;\exp\Bigl{\\{}-\frac{1}{R}\sum_{1\leq
a<b}^{N}\,|r_{a}-r_{b}|\Bigr{\\}}$ (53)

After rescaling $r_{a}\to R\,x_{a}$ the relation (51) takes the form

$2\beta\tilde{E}_{N}(R)\;I_{0}(N)\;=\;R^{-2}\,I_{\nabla}(N)\;-\;\kappa\,R^{-1}\,I_{\delta}(N)$
(54)

where

$I_{0}(N)\;=\;\int\,dx_{1}...dx_{N}\;\delta\Bigl{(}\sum_{a=1}^{N}x_{a}\Bigr{)}\,\psi^{2}(x_{1},\,...\,x_{N})$
(55)
$I_{\nabla}(N)\;=\;\int\,dx_{1}...dx_{N}\;\delta\Bigl{(}\sum_{a=1}^{N}x_{a}\Bigr{)}\,\sum_{b=1}^{N}\,\Bigl{(}\frac{\partial}{\partial
x_{a}}\psi(x_{1},\,...\,x_{N})\Bigr{)}^{2}$ (56)
$I_{\delta}(N)\;=\;\int\,dx_{1}...dx_{N}\;\delta\Bigl{(}\sum_{a=1}^{N}x_{a}\Bigr{)}\,\sum_{b\not=c}^{N}\,\delta(x_{b}-x_{c})\,\psi^{2}(x_{1},\,...\,x_{N})$
(57)

and

$\psi(x_{1},\,...\,x_{N})\;=\;\exp\Bigl{\\{}-\sum_{1\leq
a<b}^{N}\,|x_{a}-x_{b}|\Bigr{\\}}$ (58)

As the wave function $\psi(x_{1},\,...\,x_{N})$, eq.(58), exhibits no
divergences at $|x_{a}-x_{b}|\to 0$, the expression for $I_{\delta}(N)$,
eq.(57), is given in the limit $\epsilon\to 0$.

Detailed calculations of the above three factors are given in Appendix A. The
results are:

$\displaystyle I_{0}(N)$ $\displaystyle=$ $\displaystyle
2^{-(N-1)}\,\frac{1}{(N-1)!}$ (59) $\displaystyle I_{\nabla}(N)$
$\displaystyle=$ $\displaystyle\frac{1}{3}\,N(N^{2}-1)\;I_{0}(N)$ (60)
$\displaystyle I_{\delta}(N)$ $\displaystyle=$
$\displaystyle\frac{1}{3}\,N(N^{2}-1)\;I_{0}(N)$ (61)

Substituting these expressions into eq.(54) we get

$\tilde{E}_{N}(R)\;=\;\frac{1}{6\beta}\,N(N^{2}-1)\,\bigl{(}R^{-2}\;-\;\kappa\,R^{-1}\bigr{)}$
(62)

The minimum of $\tilde{E}_{N}(R)$ is achieved at $R_{*}\,=\,2/\kappa$ such
that

$\tilde{E}_{N}(R_{*})\;\equiv\;E_{*}(N)\;=\;-\frac{\kappa^{2}}{24\beta}\,(N^{3}\,-\,N)$
(63)

Substituting here $\kappa\,=\,\beta^{3}\,u$ we get

$E_{*}(N)\;=\;-\frac{1}{24}\,\beta^{5}\,u^{2}\,(N^{3}\,-\,N)$ (64)

which perfectly coincides with the exact result for the ground state energy of
$N$-particle boson system with paired attractive $\delta$ interactions (see
e.g. VD-TW-2010-2 ). In the limit $N\to\infty$ we obtain

$Z(N,t)\;\sim\;\exp\bigl{\\{}-E_{*}(N)\,t\bigr{\\}}\;\sim\;\exp\Bigl{\\{}\frac{1}{24}(\beta
u)^{2}\,(\beta N)^{3}\Bigr{\\}}$ (65)

so that according to the discussion in Section II, eqs.(19) and (27)-(29), the
free energy fluctuations scale with time as

$F\;\sim\;(\beta u)^{2/3}\,t^{1/3}\cdot f$ (66)

where the random quantity $f\sim 1$ is described by a probability distribution
function with the left tail

${\cal
P}(f\to-\infty)\;\sim\;\exp\Bigl{\\{}-(\mbox{const})\,|f|^{3/2}\Bigr{\\}}$
(67)

which also perfectly fits with well known exact results (see e.g.
LeDoussal-2010 ; VD-TW-2010-1 ).

##  V (2+1) Directed polymers: free energy scaling

According to eq.(47), to derive the scaling exponent of the free energy
fluctuations of (2+1) directed polymers we have to find the minimum of
$\tilde{E}_{N}(R)$ with respect to $R$ (which is easy), but more important, we
hate to derive the asymptotics of the ratio $I_{1}(N)/I_{0}(N)$ in the limit
$N\to\infty$ (which is not easy). We start with the treatment of the factor
$I_{1}(N)$, eq.(48):

$I_{1}(N)\;=\;\int\,d^{2}x_{2}...d^{2}x_{N}\;\delta\Bigl{(}{\bf
x_{2}}+\sum_{b=2}^{N}{\bf x}_{b}\Bigr{)}\,\exp\Bigl{\\{}-2\sum_{a=3}^{N}|{\bf
x}_{2}-{\bf x}_{a}|\;-\;2\sum_{2\leq a<b}^{N}|{\bf x}_{a}-{\bf
x}_{b}|\Bigr{\\}}$ (68)

Here and in what follows we approximate the wave function $\psi\bigl{(}{\bf
x}_{1},\,...,\,{\bf x}_{N}\bigr{)}$ by its form at scales $|{\bf x}_{a}-{\bf
x}_{b}|\gtrsim 1$, eq.(36). Note that as the integration in the vicinities of
the logarithmic singularities at $|x_{a}-x_{b}|\to\epsilon\ll 1$ is converging
its contribution is negligible compared to the one at scales $|{\bf
x}_{a}-{\bf x}_{b}|\gtrsim 1$.

It can be easily shown (see Appendix B) that at $N\gg 1$ both the typical
(average) value of the distances between the particles $\langle|{\bf
x}_{a}-{\bf x}_{b}|\rangle\;\sim\;1/N$ and the typical value of the lengths of
the vectors $\langle|{\bf x}_{a}|\rangle\;\sim\;1/N$. It makes possible to
compare the orders of magnitudes of the two factors in the exponential in the
r.h.s. of eq.(68):

$\displaystyle\sum_{a=3}^{N}|{\bf x}_{2}-{\bf
x}_{a}|\;\sim\;\frac{1}{N}\,(N-2)\;\sim\;1$ (69) $\displaystyle\sum_{2\leq
a<b}^{N}|{\bf x}_{a}-{\bf x}_{b}|\;\sim\;\frac{1}{N}\,(N-1)(N-2)\;\sim\;N$

which shows that we can neglect the first one compared to the second one.
Similarly, for the terms under the $\delta$-function of eq.(68) we have:

$\displaystyle{\bf x}_{2}\;\sim\;\frac{1}{N}$ (70)
$\displaystyle\sum_{b=2}^{N}{\bf x}_{a}\;\sim\;\frac{1}{N}\,(N-1)\;\sim\;1$

which makes possible to neglect the first term compared to the second one.

Thus, we conclude that

$I_{1}(N)\;\sim\;\int\,d^{2}x_{2}...d^{2}x_{N}\;\delta\Bigl{(}\sum_{b=2}^{N}{\bf
x}_{b}\Bigr{)}\,\exp\Bigl{\\{}-2\sum_{2\leq a<b}^{N}|{\bf x}_{a}-{\bf
x}_{b}|\Bigr{\\}}\;=\;I_{0}(N-1)$ (71)

The factor $I_{0}(N)$, eq.(38), can be represented as follows:

$\displaystyle I_{0}(N)$ $\displaystyle\sim$
$\displaystyle\int\,d^{2}x_{1}\,d^{2}x_{2}\,...\,d^{2}x_{N}\;\delta\Bigl{(}\sum_{a=1}^{N}{\bf
x}_{a}\Bigr{)}\,\exp\Bigl{\\{}-2\sum_{1\leq a<b}^{N}|{\bf x}_{a}-{\bf
x}_{b}|\Bigr{\\}}$ (72) $\displaystyle=$ $\displaystyle\int
d^{2}x_{2}...d^{2}x_{N}\;\int d^{2}x_{1}\;\delta\Bigl{(}{\bf
x}_{1}+\sum_{a=2}^{N}{\bf x}_{a}\Bigr{)}\,\exp\Bigl{\\{}-2\sum_{a=2}^{N}|{\bf
x}_{1}-{\bf x}_{a}|\;-\;2\sum_{2\leq a<b}^{N}|{\bf x}_{a}-{\bf
x}_{b}|\Bigr{\\}}$

At this stage we are going to integrate over one degree of freedom (over ${\bf
x}_{1}$) which would make possible to relate the factors $I_{0}(N)$ and
$I_{0}(N-1)$.

Let us redefine the integration variables: $\\{{\bf x}_{2},\,...\,,{\bf
x}_{N}\\}\;\to\;\\{{\bf z},\,{\bf y}_{2},\,...\,,{\bf y}_{N-1}\\}$:

$\displaystyle{\bf x}_{a}$ $\displaystyle=$ $\displaystyle{\bf z}\;+\;{\bf
y}_{a}\;\;\;\;\;(a\,=\,2,...,N-1)$ $\displaystyle{\bf x}_{N}$ $\displaystyle=$
$\displaystyle{\bf z}\;-\;\sum_{a=2}^{N-1}\,{\bf y}_{a}$ (73)

Denoting ${\bf y}_{N}\equiv-\sum_{a=2}^{N-1}\,{\bf y}_{a}$, instead of eq.(72)
we get

$I_{0}(N)\;\sim\;\int d^{2}z\int
d^{2}y_{2}...d^{2}y_{N}\,\delta\Bigl{(}\sum_{a=2}^{N}{\bf
y}_{a}\Bigr{)}\;D(N)\;\int d^{2}x_{1}\,\delta\bigl{(}{\bf x}_{1}\,+\,(N-1){\bf
z}\bigr{)}\exp\Bigl{\\{}-2\sum_{a=2}^{N}|{\bf x}_{1}-{\bf z}-{\bf
y}_{a}|\;-\;2\sum_{2\leq a<b}^{N}|{\bf y}_{a}-{\bf y}_{b}|\Bigr{\\}}$ (74)

where $D(N)$ is the Jacobian

$D(N)\;=\;\det\bigg{|}\frac{\partial\bigl{(}{\bf x}_{2},...,{\bf
x}_{N}\bigr{)}}{\partial\bigl{(}{\bf z},{\bf y}_{2},...,{\bf
y}_{N-1}\bigr{)}}\bigg{|}$ (75)

Simple calculations yield (see Appendix C)

$D(N)\;=\;(N-1)^{2}\,\Big{|}_{N\gg 1}\;\simeq\;N^{2}$ (76)

Thus, integrating over ${\bf x}_{1}$ in eq.(74), we get

$I_{0}(N)\;\sim\;N^{2}\int
d^{2}y_{2}...d^{2}y_{N}\;\delta\Bigl{(}\sum_{a=2}^{N}{\bf
y}_{a}\Bigr{)}\;\exp\Bigl{\\{}-2\sum_{2\leq a<b}^{N}|{\bf y}_{a}-{\bf
y}_{b}|\Bigr{\\}}\,\int d^{2}z\;\exp\Bigl{\\{}-2\sum_{a=2}^{N}|N{\bf z}+{\bf
y}_{a}|\Bigr{\\}}$ (77)

The above expression can be represented as follows:

$I_{0}(N)\;\sim\;N^{2}\,I_{0}(N-1)\,\int d^{2}z\,\frac{\int
d^{2}y_{2}...d^{2}y_{N}\;\delta\Bigl{(}\sum_{a=2}^{N}{\bf
y}_{a}\Bigr{)}\;\exp\Bigl{\\{}-2\sum_{2\leq a<b}^{N}|{\bf y}_{a}-{\bf
y}_{b}|\Bigr{\\}}\,\exp\Bigl{\\{}-2\sum_{a=2}^{N}|N{\bf z}+{\bf
y}_{a}|\Bigr{\\}}}{\int
d^{2}y_{2}...d^{2}y_{N}\;\delta\Bigl{(}\sum_{a=2}^{N}{\bf
y}_{a}\Bigr{)}\;\exp\Bigl{\\{}-2\sum_{2\leq a<b}^{N}|{\bf y}_{a}-{\bf
y}_{b}|\Bigr{\\}}}$ (78)

or

$I_{0}(N)\;\sim\;N^{2}\,I_{0}(N-1)\,\int
d^{2}z\;\bigg{\langle}\exp\Bigl{\\{}-2\sum_{a=2}^{N}|N{\bf z}+{\bf
y}_{a}|\Bigr{\\}}\bigg{\rangle}_{\\{{\bf y}\\}}$ (79)

where the averaging is done over $\\{{\bf y}_{2},...,{\bf y_{N}}\\}$ with the
wight $\exp\bigl{\\{}-2\sum_{2\leq a<b}^{N}|{\bf y}_{a}-{\bf
y}_{b}|\bigr{\\}}$. Redefining ${\bf z}=\frac{1}{2N^{2}}{\bf u}$ and ${\bf
y}_{a}=\frac{1}{2N}{\bf v}_{a}$ we get

$I_{0}(N)\;\sim\;\frac{1}{4N^{2}}\,I_{0}(N-1)\,\int
d^{2}u\;\bigg{\langle}\exp\Bigl{\\{}-\frac{1}{N}\sum_{a=2}^{N}|{\bf u}+{\bf
v}_{a}|\Bigr{\\}}\bigg{\rangle}_{\\{{\bf v}\\}}$ (80)

where the averaging is done over $\\{{\bf v}_{2},...,{\bf v_{N}}\\}$ with the
wight $\exp\bigl{\\{}-\frac{1}{N}\sum_{2\leq a<b}^{N}|{\bf v}_{a}-{\bf
v}_{b}|\bigr{\\}}$. As the typical value of ${\bf y}_{a}$ (defined by the
distribution $\exp\bigl{\\{}-2\sum_{2\leq a<b}^{N}|{\bf y}_{a}-{\bf
y}_{b}|\bigr{\\}}$) is $\sim N^{-1}$ (see Appendix B), the typical value of
the vectors ${\bf v}_{a}$ is $\sim 1$. Thus, we can estimate

$\Phi({\bf
u})\equiv\bigg{\langle}\exp\Bigl{\\{}-\frac{1}{N}\sum_{a=2}^{N}|{\bf u}+{\bf
v}_{a}|\Bigr{\\}}\bigg{\rangle}_{\\{{\bf
v}\\}}\;\sim\;\left\\{\begin{array}[]{ll}\bigg{\langle}\exp\Bigl{\\{}-\frac{1}{N}\sum_{a=2}^{N}|{\bf
v}_{a}|\Bigr{\\}}\bigg{\rangle}_{\\{{\bf v}\\}}\,=\,C_{0},\sim
1\;,\;\;\;\;\mbox{at}\;|{\bf u}|\,\ll 1\\\ \\\
\bigg{\langle}\exp\Bigl{\\{}-\frac{1}{N}\sum_{a=2}^{N}|{\bf
u}|\Bigr{\\}}\bigg{\rangle}_{\\{{\bf v}\\}}\,\simeq\exp\bigl{\\{}-|{\bf
u}|\bigr{\\}},\;\;\;\;\;\;\;\mbox{at}\;|{\bf u}|\,\gg 1\end{array}\right.$
(81)

which means, according to eq.(80), that

$I_{0}(N)\;\sim\;\frac{1}{4N^{2}}\,I_{0}(N-1)\,\int d^{2}u\;\Phi({\bf u})\;\
\sim\;N^{-2}\,I_{0}(N-1)$ (82)

or, according to eq.(71)

$I_{0}(N-1)\;\sim\;I_{1}(N)\;\sim\;N^{2}\,I_{0}(N)$ (83)

Substituting this into eq.(47), we get

$\tilde{E}_{N}(R)\;\sim\;\frac{1}{\beta}\,N^{4}\,\,\Bigl{[}R^{-2}\,\ln\Bigl{(}\frac{R}{\epsilon}\Bigr{)}\;-\;\kappa\,R^{-2}\,\ln^{2}\Bigl{(}\frac{R}{\epsilon}\Bigr{)}\Bigr{]}$
(84)

This function of $R$ has the minimum at

$\ln\Bigl{(}\frac{R_{*}}{\epsilon}\Bigr{)}\;=\;\frac{1}{2\kappa}\,\bigl{(}1\,+\,\kappa\,+\,\sqrt{1\,+\,\kappa^{2}}\bigr{)}\,\Big{|}_{\kappa\gg
1}\;\simeq\;1$ (85)

and its value at the minimum

$\tilde{E}_{N}(R_{*})\,\equiv\,E_{*}(N)\,\sim\,-\frac{\kappa}{\beta\epsilon^{2}}\,N^{4}\;=\;-\frac{u}{({\beta\,\epsilon})^{2}}\,\bigl{(}\beta\,N\bigr{)}^{4}$
(86)

Correspondingly, for the replica partition function we get

$Z(N,t)\;\sim\;\exp\bigl{\\{}-E_{*}(N)\,t\bigr{\\}}\;\sim\;\exp\Bigl{\\{}\frac{u}{({\beta\,\epsilon})^{2}}\,\bigl{(}\beta\,N\bigr{)}^{4}\,t\Bigr{\\}}$
(87)

Thus, according to eqs.(19) and (27)-(29), the free energy fluctuations in the
considered system scale with time as

$F\;\sim\;\Bigl{(}\frac{\sqrt{u}}{\beta\,\epsilon}\Bigr{)}^{1/2}\,t^{1/4}\cdot
f$ (88)

where the random quantity $f\sim 1$ is described by a probability distribution
function with the left tail

${\cal
P}(f\to-\infty)\;\sim\;\exp\Bigl{\\{}-(\mbox{const})\,|f|^{4/3}\Bigr{\\}}$
(89)

The above eqs.(88) and (89) constitute the main result of the present
research.

##  VI Discussion

In this paper we argue that at least in the low temperature limit the scaling
exponent of the free energy fluctuations of $(2+1)$ directed polymers is equal
to $\theta=1/4$ which close but not equal to the value $0.241$ obtained in
numerical simulations. I would suggest that this discrepancy could be due to
the presence of a logarithmic prefactor which in numerical simulations may
distort the ”pure” scaling behavior with $\theta=1/4$ and which is out of
reach of the present analytical method.

It is worthing to stress that according to eq.(88), the prefactor in the time
scaling of the free energy fluctuation explicitly depend on the size
$\epsilon$ of the spatial correlations of the random potential, eq.(4).
Moreover, we see that the limit $\epsilon\to 0$ (when these correlations turn
into the true $\delta$-function) do not exist here (the prefactor is
divergent). In other words, one can not consider the (2+1) directed polymers
with $\delta$-correlated random potential because such system is pathological.
This is in drastic difference with (1+1) directed polymers where it is just
for the $\delta$-correlated random potential that the exact solution has been
derived.

Of course, present research is based on the calculations which are far from
being rigorous, and correspondingly it leaves many open questions. The main
problem is that the ground state wave function for the considered two-
dimensional $N$-particle boson system with short-range attractive interactions
(which should be the solution of eq.(13)) is not known. The key assumption of
the present approach is that this wave function can be approximated by the
heuristic structure given in eqs.(34)-(35), which is a straightforward
generalization of the two-particle solution for the $N$-particle system and
which contains one (spatial scale) adjustment parameter. It is evident that
such structure of the wave function is not correct for an arbitrary (finite)
value of $N$, but on the other hand it seems to be a good approximation in the
limit $N\to\infty$. To what extend the use of such approximation is essential
for the declared result $\theta=1/4$ is not clear.

Another key point of the present approach is the hypothesis that the free
energy fluctuations of the considered system are described by a universal
probability distribution function. This is known to take place in (1+1)
directed polymers where the exact solution is available, but it is not
necessary guaranteed for the considered system. If this hypothesis is correct
then for obtaining the value of the scaling exponent it is sufficient to
derive the left tail of the free energy distribution function (that is what
has been done in this paper). If not, then the knowledge of this left tail
tells us just nothing.

One more problem is the evident dependence of the scaling properties of the
considered system on the temperature. The present study was devoted to the low
temperature limit when dimensionless parameter $\kappa=\beta^{3}\,u\;\gg\;1$
(see eqs.(85)-(86)) and in this case we get $\theta=1/4$. Unfortunately in the
framework of the present approach one can not study the high temperature limit
(when $\kappa=\beta^{3}\,u\;\ll\;1$) as in this case the minimum of the
function $\tilde{E}_{N}(R)$, eq.(84), becomes almost ”flat” and
correspondingly the value of $R_{*}$ becomes ill defined. On the other hand,
previous analytical study of the high temperature limit (using completely
different method) (2+1)-highT  yields the result $\theta=1/2$. If these two
results are both correct then we would have to conclude that unlike (1+1)
directed polymers the value of the scaling exponent in the present system is
non-universal being temperature dependent.

Finally, like in the case of (1+1) directed polymers the zero temperature
limit ($\beta\to\infty$) in the considered (2+1) system remains non accessible
in the framework of the usual continuous-space and replica symmetric approach.
Either we have to consider the lattice version of this model or we have to
introduce some kind of a replica symmetry breaking ansatz zero-T .

In any case further systematic study (both theoretical an numerical) of this
system would be highly welcomed.

Appendix A

For the calculations of the integrals in eqs.(55)-(57) due to the symmetry of
the wave function $\psi(x_{1},\,...\,x_{N})$ we can order the particle’s
positions, e.g. as follows:

$\\{x_{1},x_{2},...,x_{N}\\}\;\to\;\\{-\infty<x_{1}\leq x_{2}\leq...\leq
x_{N}<+\infty\\}$ (A.1)

In this case

$\int\,dx_{1}\,...\,dx_{N}\;\to\;N!\underbrace{\int
dx_{1}\,...\,dx_{N}}_{-\infty<x_{1}\leq...\leq x_{N}<+\infty}$ (A.2)

and

$\sum_{1\leq
a<b}^{N}\,|x_{a}-x_{b}|\;=\;-\sum_{k=1}^{N}\bigl{(}N-2k+1\bigr{)}\,x_{k}$
(A.3)

so that the wave function takes the form

$\psi(x_{1},\,...\,x_{N})\;=\;\exp\Bigl{\\{}\sum_{k=1}^{N}\,\mu_{k}\,x_{k}\Bigr{\\}}$
(A.4)

where

$\mu_{k}\;=\;\bigl{(}N-2k+1\bigr{)}$ (A.5)

For the factor $I_{0}(N)$, eq.(55), we get the following integral
representation

$I_{0}(N)\;=\;2\,N!\,\int_{-\infty}^{+\infty}\frac{dp}{2\pi}\int_{-\infty}^{+\infty}dx_{N}\int_{-\infty}^{x_{N}}dx_{N-1}\;...\;\int_{-\infty}^{x_{3}}dx_{2}\int_{-\infty}^{x_{2}}dx_{1}\exp\Bigl{\\{}2\sum_{k=1}^{N}\bigl{(}\mu_{k}+ip\bigr{)}x_{k}\Bigr{\\}}$
(A.6)

Integrating one after another over $x_{1},x_{2},...,x_{N-1}$ we obtain

$I_{0}(N)\;=\;2^{-(N-1)}\,N!\,\int_{-\infty}^{+\infty}\frac{dp}{2\pi}\int_{-\infty}^{+\infty}dx_{N}\frac{\exp\Bigl{\\{}\Bigl{(}\sum_{k=1}^{N}\mu_{k}\;+\;iNp\Bigr{)}\,x_{N}\Bigr{\\}}}{(\mu_{1}+ip)(\mu_{1}+\mu_{2}+2ip)\,...\,(\mu_{1}+...+\mu_{N-1}+i(N-1)p)}$
(A.7)

Taking into account that $\sum_{k=1}^{N}\mu_{k}\,=\,0$ we eventually find

$\displaystyle I_{0}(N)$ $\displaystyle=$ $\displaystyle
2^{-(N-1)}\,N!\,\int_{-\infty}^{+\infty}dp\frac{\delta(Np)}{(\mu_{1}+ip)(\mu_{1}+\mu_{2}+2ip)\,...\,(\mu_{1}+...+\mu_{N-1}+i(N-1)p)}$
(A.8) $\displaystyle=$ $\displaystyle
2^{-(N-1)}\,N!\,\frac{1}{N}\,\frac{1}{\mu_{1}(\mu_{1}+\mu_{2})\,...\,(\mu_{1}+...+\mu_{N-1})}$
$\displaystyle=$ $\displaystyle
2^{-(N-1)}\,(N-1)!\,\frac{1}{(N-1)(2N-4)(3N-9)\,...\,\bigl{[}(N-1)-(N-1)(N-1)\bigr{]}}$
$\displaystyle=$ $\displaystyle
2^{-(N-1)}\,(N-1)!\,\frac{1}{\bigl{[}(N-1)!\bigr{]}^{2}}$ $\displaystyle=$
$\displaystyle\frac{2^{-(N-1)}}{(N-1)!}$

In a similar way for the factor $I_{\nabla}(N)$, eq.(56), we obtain

$\displaystyle I_{\nabla}(N)$ $\displaystyle=$ $\displaystyle
2\,N!\,\sum_{k=1}^{N}\mu_{k}^{2}\;\int_{-\infty}^{+\infty}\frac{dp}{2\pi}\int_{-\infty}^{+\infty}dx_{N}\int_{-\infty}^{x_{N}}dx_{N-1}\;...\;\int_{-\infty}^{x_{2}}dx_{1}\;\exp\Bigl{\\{}2\sum_{k=1}^{N}\bigl{(}\mu_{k}+ip\bigr{)}x_{k}\Bigr{\\}}$
(A.9) $\displaystyle=$ $\displaystyle\sum_{k=1}^{N}(N-2k+1)^{2}\;I_{0}(N)$
$\displaystyle=$ $\displaystyle\frac{1}{3}\,N(N^{2}-1)\;I_{0}(N)$

The calculations of $I_{\delta}(N)$, eq.(57), are slightly more cumbersome:

$\displaystyle I_{\delta}(N)$ $\displaystyle=$ $\displaystyle
N!\;\sum_{k=1}^{N-1}\underbrace{\int
dx_{1}\,...\,dx_{N}}_{-\infty<x_{1}\leq...\leq
x_{N}<+\infty}\delta\bigl{(}x_{k}-x_{k+1}\bigr{)}\,\delta\Bigl{(}\sum_{l=1}^{N}x_{l}\Bigr{)}\,\exp\Bigl{\\{}2\sum_{n=1}^{N}\mu_{n}\,x_{n}\Bigr{\\}}$
(A.10) $\displaystyle=$ $\displaystyle
2^{-(N-2)}\,N!\,\int_{-\infty}^{+\infty}\frac{dp}{2\pi}\sum_{k=1}^{N-1}\underbrace{\int
dx_{1}\,...\,dx_{k}\,\int dx_{k+2}\,...\,dx_{N}}_{-\infty<x_{1}\leq...\leq
x_{k}\leq x_{k+2}\leq...\leq x_{N}<+\infty}\;\times$ $\displaystyle\times$
$\displaystyle\exp\Bigl{\\{}\sum_{n=1}^{k-1}\bigl{(}\mu_{n}+ip\bigr{)}\,x_{n}\,+\,\bigl{(}\mu_{k}+\mu_{k+1}+2ip\bigr{)}\,x_{k}\,+\,\sum_{n=k+2}^{N}\bigl{(}\mu_{n}+ip\bigr{)}\,x_{n}\Bigr{\\}}$

Integrating over $\\{x_{1},...,x_{k},x_{k+2},...,x_{N}\\}$ (s.f.
eqs.(A.6)-(A.8)) and taking into account that $\sum_{n=1}^{N}\mu_{n}=0$, we
get

$\displaystyle I_{\delta}(N)$ $\displaystyle=$ $\displaystyle
2^{-(N-2)}\,N!\,\int_{-\infty}^{+\infty}dp\sum_{k=1}^{N-1}\;\frac{\delta\bigl{(}N\,p\bigr{)}}{\mu_{1}\,(\mu_{1}+\mu_{2})\,...\,(\mu_{1}+...+\mu_{k-1})\times(\mu_{1}+...+\mu_{k+1})\,...\,(\mu_{1}+...+\mu_{N-1})}$
(A.11) $\displaystyle=$ $\displaystyle
2^{-(N-2)}\,(N-1)!\sum_{k=1}^{N-1}\;\frac{(\mu_{1}+\mu_{2}+...+\mu_{k})}{\mu_{1}\,(\mu_{1}+\mu_{2})\,...\,(\mu_{1}+...+\mu_{N-1})}$
$\displaystyle=$ $\displaystyle
2^{-(N-2)}\,(N-1)!\,\frac{1}{\bigl{(}(N-1)!\bigr{)}^{2}}\,\sum_{k=1}^{N-1}\sum_{n=1}^{k}\,(N-2n+1)$
$\displaystyle=$ $\displaystyle
2\,I_{0}(N)\,\sum_{k=1}^{N-1}\,(N-k)\,(N-2k+1)$ $\displaystyle=$
$\displaystyle\frac{1}{3}\,N(N^{2}-1)\,I_{0}(N)$

Appendix B

The average distance between particles can be computed as follows

$\langle|\Delta{\bf x}|\rangle\;=\;\frac{2}{N(N-1)}\sum_{a<b}^{N}\langle|{\bf
x}_{a}-{\bf x}_{b}|\rangle\;=\;\frac{2\int
d^{2}x_{1}\,...\,d^{2}x_{N}\;\delta\Bigl{(}\sum_{a=1}^{N}{\bf
x}_{a}\Bigr{)}\,\sum_{a<b}^{N}|{\bf x}_{a}-{\bf
x}_{b}|\;\exp\Bigl{\\{}-2\sum_{1\leq a<b}^{N}|{\bf x}_{a}-{\bf
x}_{b}|\Bigr{\\}}}{N(N-1)\;\int
d^{2}x_{1}\,...\,d^{2}x_{N}\;\delta\Bigl{(}\sum_{a=1}^{N}{\bf
x}_{a}\Bigr{)}\,\exp\Bigl{\\{}-2\sum_{1\leq a<b}^{N}|{\bf x}_{a}-{\bf
x}_{b}|\Bigr{\\}}}$ (B.1)

or

$\langle|\Delta{\bf
x}|\rangle\;=\;-\frac{2}{N(N-1)}\;\frac{\partial}{\partial\lambda}\;\ln\biggl{[}\int
d^{2}x_{1}\,...\,d^{2}x_{N}\;\delta\Bigl{(}\sum_{a=1}^{N}{\bf
x}_{a}\Bigr{)}\,\exp\Bigl{\\{}-\lambda\sum_{1\leq a<b}^{N}|{\bf x}_{a}-{\bf
x}_{b}|\Bigr{\\}}\biggr{]}\,\bigg{|}_{\lambda=2}$ (B.2)

After rescaling ${\bf x}_{a}\to\lambda^{-1}\,{\bf x}_{a}$ we get

$\displaystyle\langle|\Delta{\bf x}|\rangle$ $\displaystyle=$
$\displaystyle-\frac{2}{N(N-1)}\;\frac{\partial}{\partial\lambda}\;\ln\biggl{[}\lambda^{-2(N-1)}\int
d^{2}x_{1}\,...\,d^{2}x_{N}\;\delta\Bigl{(}\sum_{a=1}^{N}{\bf
x}_{a}\Bigr{)}\,\exp\Bigl{\\{}-\sum_{1\leq a<b}^{N}|{\bf x}_{a}-{\bf
x}_{b}|\Bigr{\\}}\biggr{]}\,\bigg{|}_{\lambda=2}$ (B.3) $\displaystyle=$
$\displaystyle-\frac{2}{N(N-1)}\;\frac{\partial}{\partial\lambda}\;\biggl{\\{}-2(N-1)\ln(\lambda)\;+\;\ln\biggl{[}\int
d^{2}x_{1}\,...\,d^{2}x_{N}\;\delta\Bigl{(}\sum_{a=1}^{N}{\bf
x}_{a}\Bigr{)}\,\exp\Bigl{\\{}-\sum_{1\leq a<b}^{N}|{\bf x}_{a}-{\bf
x}_{b}|\Bigr{\\}}\biggr{]}\biggr{\\}}\,\bigg{|}_{\lambda=2}$ $\displaystyle=$
$\displaystyle\frac{2}{N(N-1)}\,2(N-1)\,\frac{1}{\lambda}\,\Big{|}_{\lambda=2}\;=\;\frac{2}{N}$
$\displaystyle\sim$ $\displaystyle\,\frac{1}{N}$

As for any two particles $|{\bf x}_{a}-{\bf x}_{b}|\,\sim\,N^{-1}$, the square
of this difference $\bigl{(}{\bf x}_{a}-{\bf
x}_{b}\bigr{)}^{2}\,\sim\,N^{-2}$, or

$\frac{1}{N(N-1)}\sum_{a,b}^{N}\bigl{(}{\bf x}_{a}-{\bf
x}_{b}\bigr{)}^{2}\;=\;\frac{1}{N(N-1)}\Bigl{[}2N\sum_{a=1}^{N}{\bf
x}_{a}^{2}\;-\;2\,\Bigl{(}\sum_{a=1}^{N}{\bf
x}_{a}\Bigr{)}^{2}\Bigr{]}\;\sim\;N^{-2}$ (B.4)

Taking into account that $\sum_{a=1}^{N}{\bf x}_{a}\;=\;0$ we obtain

$\frac{2}{(N-1)}\,\sum_{a=1}^{N}{\bf x}_{a}^{2}\;\sim\;N^{-2}$ (B.5)

which means that ${\bf x}_{a}^{2}\,\sim\,N^{-2}$ and $|{\bf
x}_{a}|\,\sim\,N^{-1}$.

Appendix C

The Jacobian of the linear transformation

$\displaystyle{\bf x}_{a}$ $\displaystyle=$ $\displaystyle{\bf z}\;+\;{\bf
y}_{a}\;\;\;\;\;(a\,=\,2,...,N-1)$ $\displaystyle{\bf x}_{N}$ $\displaystyle=$
$\displaystyle{\bf z}\;-\;\sum_{a=2}^{N-1}\,{\bf y}_{a}$ (C.1)

is

$D(N)\;=\;\det\bigg{|}\frac{\partial\bigl{(}{\bf x}_{2},...,{\bf
x}_{N}\bigr{)}}{\partial\bigl{(}{\bf z},{\bf y}_{2},...,{\bf
y}_{N-1}\bigr{)}}\bigg{|}$ (C.2)

In the Cartesian vector representation ${\bf x}_{a}=(p_{a},q_{a})\,;\;\;{\bf
y}_{a}=(s_{a},t_{a})\,;\;\;{\bf z}_{a}=(u,v)$ the above linear transformation
reads

$\displaystyle p_{a}$ $\displaystyle=$ $\displaystyle
u\;+\;s_{a}\;\;\;\;\;(a\,=\,2,...,N-1)$ $\displaystyle p_{N}$ $\displaystyle=$
$\displaystyle u\;-\;\sum_{a=2}^{N-1}\,s_{a}$ (C.3) $\displaystyle q_{a}$
$\displaystyle=$ $\displaystyle v\;+\;t_{a}\;\;\;\;\;(a\,=\,2,...,N-1)$
$\displaystyle q_{N}$ $\displaystyle=$ $\displaystyle
v\;-\;\sum_{a=2}^{N-1}\,t_{a}$ (C.4)

so that

$D(N)\;=\;\det\bigg{|}\frac{\partial\bigl{(}p_{2},...,p_{N}\bigr{)}}{\partial\bigl{(}u,s_{2},...,s_{N-1}\bigr{)}}\bigg{|}\,\times\,\det\bigg{|}\frac{\partial\bigl{(}q_{2},...,q_{N}\bigr{)}}{\partial\bigl{(}v,t_{2},...,t_{N-1}\bigr{)}}\bigg{|}$
(C.5)

Explicitly,

$\det\bigg{|}\frac{\partial\bigl{(}p_{2},...,p_{N}\bigr{)}}{\partial\bigl{(}u,s_{2},...,s_{N-1}\bigr{)}}\bigg{|}\;=\;\det\,\begin{tabular}[]{|c|c|c|c|c|}\hline\cr$\frac{\partial
p_{2}}{\partial u}$&$\frac{\partial p_{3}}{\partial u}$&$\frac{\partial
p_{4}}{\partial u}$&...&$\frac{\partial p_{N}}{\partial u}$\\\
\hline\cr$\frac{\partial p_{2}}{\partial s_{2}}$&$\frac{\partial
p_{3}}{\partial s_{2}}$&$\frac{\partial p_{4}}{\partial s_{2}}$&\;
...&$\frac{\partial p_{N}}{\partial s_{2}}$\\\ \hline\cr$\frac{\partial
p_{2}}{\partial s_{3}}$&$\frac{\partial p_{3}}{\partial
s_{3}}$&$\frac{\partial p_{4}}{\partial s_{3}}$&...&$\frac{\partial
p_{N}}{\partial s_{3}}$\\\ \hline\cr\vrule\lx@intercol\hfil. . . . . . . . . .
. .\hfil\lx@intercol\vrule\lx@intercol \\\ \hline\cr$\frac{\partial
p_{2}}{\partial s_{N-1}}$&$\frac{\partial p_{3}}{\partial
s_{N-1}}$&$\frac{\partial p_{4}}{\partial s_{N-1}}$&...&$\frac{\partial
p_{N}}{\partial s_{N-1}}$\\\ \hline\cr\end{tabular}$ (C.6)

Simple calculations yield

$\det\bigg{|}\frac{\partial\bigl{(}p_{2},...,p_{N}\bigr{)}}{\partial\bigl{(}u,s_{2},...,s_{N-1}\bigr{)}}\bigg{|}\;=\;\det\,\begin{tabular}[]{|c|c|c|c|c|}\hline\cr$\;\;1\;\;$&$\;\;1\;\;$&$\;\;1\;\;$&...&$1$\\\
\hline\cr$1$&$0$&$0$&\; ...&$-1$\\\ \hline\cr$0$&$1$&$0$&...&$-1$\\\
\hline\cr\vrule\lx@intercol\hfil. . . . . . . . . . .
.\hfil\lx@intercol\vrule\lx@intercol \\\ \hline\cr$0$&$0$&$0$&...&$-1$\\\
\hline\cr\end{tabular}\;=\;(-1)^{N}\,(N-1)$ (C.7)

Evidently, one gets the same result for the second determinant in the r.h.s of
eq.(C.5). Thus, we obtain

$D(N)\;=\;(N-1)^{2}\Big{|}_{N\gg 1}\;\simeq\;N^{2}$ (C.8)

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