The non-linear supersymmetric hyperbolic sigma model on a complete graph with
hierarchical interactions
Margherita Disertori111Institute for Applied Mathematics & Hausdorff Center
for Mathematics, University of Bonn,
Endenicher Allee 60, D-53115 Bonn, Germany. E-mail<EMAIL_ADDRESS>Franz Merkl 222Mathematisches Institut, Ludwig-Maximilians-Universität
München, Theresienstr. 39, D-80333 Munich, Germany. E-mail<EMAIL_ADDRESS>Silke W.W. Rolles333 Zentrum Mathematik, Bereich M5, Technische Universität
München, Boltzmannstr. 3, D-85748 Garching bei München, Germany. E-mail:
<EMAIL_ADDRESS>

###### Abstract

We study the non-linear supersymmetric hyperbolic sigma model $H^{2|2}$ on a
complete graph with hierarchical interactions. For interactions which do not
decrease too fast in the hierarchical distance, we prove tightness of certain
spin variables in horospherical coordinates, uniformly in the pinning and in
the size of the graph. The proof relies on a reduction to an effective
$H^{2|2}$ model; its size is logarithmic in the size of the original model.
4442020 Mathematics Subject Classification: Primary: 82B20, Secondary: 60G60.
555Keywords: non-linear supersymmetric hyperbolic sigma model, hierarchical
interactions, supersymmetry, effective model.

##  1 Introduction

###  1.1 History of the $H^{2|2}$-model and related models

The supersymmetric hyperbolic sigma model, $H^{2|2}$-model for short, was
introduced by Zirnbauer in [Zir91] as a “toy model for studying diffusion and
localization in disordered one-electron systems”. It is a supersymmetric
variant of a statistical mechanics model on a finite graph. The model deals
with spin variables taking values in the supermanifold $H^{2|2}$ with two
Grassmann components over the hyperboloid
$H^{2}=\\{(x,y,z)\in{\mathbb{R}}^{3}:x^{2}+y^{2}-z^{2}=-1,z>0\\}$. The model
has supersymmetries, which extend the generators of the Lorentz group acting
on $H^{2}$. In [DSZ10], Disertori, Spencer, and Zirnbauer examine this model
over boxes in ${\mathbb{Z}}^{d}$, $d\geq 3$. For sufficiently small
temperature, they derive moment estimates and conclude that the spins are
aligned with high probability. For high temperature in any dimension $d$,
Disertori and Spencer [DS10] show exponential decay of the $y$-$y$-two-point
function for the $H^{2|2}$-model. The result holds for any temperature in
dimension one. In two dimensions, polynomial decay of the correlations was
shown for all temperatures independently by Kozma and Peled [KP21] and Sabot
[Sab21], using different tools.

Surprisingly, although the model was originally designed to describe condensed
matter systems, there is a close connection with vertex-reinforced jump
processes as was observed by Sabot and Tarrès [ST15]. All spin variables of
the $H^{2|2}$-model written in horospherical coordinates have a probabilistic
interpretation in terms of the vertex-reinforced jump process as is worked out
in [MRT19].

There is another surprising connection of the $H^{2|2}$-model to certain
random Schrödinger operators discovered by Sabot, Tarrès, and Zeng in [STZ17]
and [SZ19]. For the analysis of these random Schrödinger operators, certain
martingales derived from the spin variables of the $H^{2|2}$-model play an
essential role. These martingales are generalized in [DMR17] and
supersymmetric variants of them are described in [DMR19].

The classical isomorphism theorems relating the local times of a continuous-
time random walk to the square of a Gaussian free field are generalized by
Bauerschmidt, Helmuth, and Swan in [BHS21] to spin systems taking values in
hyperbolic and spherical geometries.

There are generalizations of the $H^{2|2}$-model, called $H^{p|2q}$-models,
which deal with $p$ real-valued spin variables and $2q$ Grassmann variables
per vertex. For $p=2$, these models are examined first by Crawford in [Cra21].
The case of the $H^{0|2}$-model, which describes the arboreal gas
(configurations of unrooted spanning forests), and the closely related
$H^{2|4}$-model are examined by Bauerschmidt, Crawford, Helmuth, and Swan in
[BCHS21] and [BCH21].

In particular, the $H^{0|2}$-model can be represented by a purely Fermionic
model with a Gaussian measure perturbed by a short range interaction. This
allows to apply a rigorous renormalization group analysis in [BCH21]. It is
not clear how to extend such an analysis to the $H^{2|2}$-model, the main
obstruction being the hyperbolic structure and the presence of long range
interactions. If one considers the corresponding vertex-reinforced jump
process on hierarchical lattices, informal computations by Abdesselam,
Brydges, Helmuth, Lupu, Sabot, Swan, and Zeng done during an AIM
workshop666AIM Workshop on Self-interacting processes, supersymmetry, and
Bayesian statistics organized by P. Diaconis, M. Disertori, C. Sabot, and P.
Tarrès, 2019,
https://aimath.org/pastworkshops/selfsuperbayes.html suggest, for a special
choice of the pinning and the interactions, the existence of a renormalization
semigroup operation, which leaves the process invariant under appropriate
scaling.

A recent survey on the subject is given by Bauerschmidt and Helmuth in [BH21].

###  1.2 The $H^{2|2}$-model on general graphs

#### Overview of the paper.

In this paper, we deal with the $H^{2|2}$-model on a complete graph of size
$2^{N}$ with interactions depending on the hierarchical distance in a binary
tree. For some observables this model can be reduced to an effective
$H^{2|2}$-model with only $O(N)$ vertices. This reduction is described in
Theorem 2.2 below. As an application of this reduction, we prove tightness of
differences of certain spin variables in the original model, provided that the
interactions do not decay too fast with the hierarchical distance. This is
made precise in Theorem 2.1.

#### Introduction of the model.

Let $\Lambda$ be a finite non-empty set. We view it as the vertex set of a
complete undirected graph with edge set
${\mathcal{E}}_{\Lambda}:=\\{\\{i,j\\}:i,j\in\Lambda\\}$. To construct the
supermanifold $H^{2|2}$, we introduce a Grassmann algebra
$\mathcal{A}=\mathcal{A}_{0}\oplus\mathcal{A}_{1}$ with the even (commuting)
subalgebra $\mathcal{A}_{0}$ and the odd (anticommuting) subspace
$\mathcal{A}_{1}$; see [Swa20] for more details. To keep the exposition self-
contained, we also give some details in the appendix. We assume that
$\mathcal{A}$ is large enough so that we can associate to each vertex
$i\in\Lambda$ three variables $x_{i},y_{i},z_{i}$ taking values in
$\mathcal{A}_{0}$ and linearly independently two odd (Grassmann) variables
$\xi_{i},\eta_{i}\in\mathcal{A}_{1}$. Each even element $x\in\mathcal{A}_{0}$
can be decomposed in a unique way in a real part
$\operatorname{body}(x)\in{\mathbb{R}}$ and a nilpotent part
$x-\operatorname{body}(x)$. We define
$H^{3|2}:=\mathcal{A}_{0}^{3}\times\mathcal{A}_{1}^{2}$ and endow it with the
inner product

$\displaystyle\left\langle{v,v^{\prime}}\right\rangle:=xx^{\prime}+yy^{\prime}-zz^{\prime}+\xi\eta^{\prime}-\eta\xi^{\prime}$
(1.1)

for
$v=(x,y,z,\xi,\eta),v^{\prime}=(x^{\prime},y^{\prime},z^{\prime},\xi^{\prime},\eta^{\prime})\in
H^{3|2}$. The supermanifold $H^{2|2}$ can now be defined by

$\displaystyle H^{2|2}=$ $\displaystyle\\{v=(x,y,z,\xi,\eta)\in
H^{3|2}:\,\left\langle{v,v}\right\rangle=-1,\operatorname{body}(z)>0\\}$
$\displaystyle=$ $\displaystyle\\{(x,y,z,\xi,\eta)\in
H^{3|2}:\,x^{2}+y^{2}-z^{2}+2\xi\eta=-1,\operatorname{body}(z)>0\\}.$ (1.2)

For any smooth function $f:{\mathbb{R}}^{k}\to{\mathbb{R}}$ the corresponding
extension to a superfunction $f:\mathcal{A}_{0}^{k}\to\mathcal{A}_{0}$ is
denoted by the same symbol $f$ and is constructed by a Taylor expansion in the
nilpotent parts.

The superintegration form $\mathcal{D}v$ on $H^{2|2}$ is defined by

$\displaystyle
f\mapsto\int_{H^{2|2}}\mathcal{D}vf(v):=\frac{1}{2\pi}\int_{{\mathbb{R}}^{2}}dx\,dy\,\partial_{\xi}\partial_{\eta}\left(\frac{1}{z}f(x,y,z,\xi,\eta)\right),$
(1.3)

where

$\displaystyle
z=\sqrt{1+x^{2}+y^{2}+2\xi\eta}=\sqrt{1+x^{2}+y^{2}}+\frac{\xi\eta}{\sqrt{1+x^{2}+y^{2}}}$
(1.4)

according to the constraint $v\in H^{2|2}$. Here, $f$ is any superfunction
decaying sufficiently fast to make the integral well-defined.

We abbreviate

$\displaystyle o:=(0,0,1,0,0)\in H^{2|2}.$ (1.5)

We take a weight function $W:{\mathcal{E}}_{\Lambda}\to[0,\infty)$,
$\\{i,j\\}\mapsto W_{ij}=W_{ji}$ with $W_{ii}=0$ for all $i\in\Lambda$ and
consider the set of edges

$\displaystyle{\mathcal{E}}_{\Lambda,W}:=\\{\\{i,j\\}\in{\mathcal{E}}_{\Lambda}:W_{ij}>0\\}.$
(1.6)

Furthermore, we take a pinning strength function $h:\Lambda\to[0,\infty)$,
$i\mapsto h_{i}$. Throughout the paper, we assume that it fulfills $h_{i}>0$
for at least one $i$ in every connected component of the graph
$(\Lambda,{\mathcal{E}}_{\Lambda,W})$. This connectivity assumption guarantees
that the following $H^{2|2}$-model is well-defined: its action with couplings
$W$ and pinning $h$ is defined for
$v_{\Lambda}:=(v_{j})_{j\in\Lambda}\in(H^{2|2})^{\Lambda}$ by

$\displaystyle S_{W,h}^{\Lambda}(v_{\Lambda}):=$
$\displaystyle-\frac{1}{2}\sum_{i,j\in\Lambda}W_{ij}\left\langle{v_{i}-v_{j},v_{i}-v_{j}}\right\rangle+\sum_{i\in\Lambda}h_{i}(1+\left\langle{v_{i},o}\right\rangle)$
$\displaystyle=$
$\displaystyle\sum_{i,j\in\Lambda}W_{ij}(1+\left\langle{v_{i},v_{j}}\right\rangle)+\sum_{i\in\Lambda}h_{i}(1+\left\langle{v_{i},o}\right\rangle).$
(1.7)

The $H^{2|2}$-model over $\Lambda$ is given by the superintegration form

$\displaystyle f\mapsto
E_{W,h}^{\Lambda}[f(v_{\Lambda})]:=\int_{(H^{2|2})^{\Lambda}}\mathcal{D}v_{\Lambda}\left(e^{S^{\Lambda}_{W,h}(v_{\Lambda})}f(v_{\Lambda})\right)$
(1.8)

with the abbreviation

$\displaystyle\mathcal{D}v_{\Lambda}:=\prod_{j\in\Lambda}\mathcal{D}v_{j}.$
(1.9)

By supersymmetry, $E^{\Lambda}_{W,h}[1]=1$ for the constant function
$f(v_{\Lambda})\equiv 1$, see formula (5.1) in [DSZ10].

#### Horospherical coordinates.

For any vertex $i\in\Lambda$ and
$v_{i}=(x_{i},y_{i},z_{i},\xi_{i},\eta_{i})\in H^{2|2}$, we introduce
horospherical coordinates consisting of even components $u_{i},s_{i}$ and
Grassmann variables $\overline{\psi}_{i}$, $\psi_{i}$ as follows, cf. Section
2.2 in [DSZ10]:

$\displaystyle u_{i}=\log(x_{i}+z_{i}),\quad
s_{i}=\frac{y_{i}}{x_{i}+z_{i}},\quad\overline{\psi}_{i}=\frac{\xi_{i}}{x_{i}+z_{i}},\quad\psi_{i}=\frac{\eta_{i}}{x_{i}+z_{i}}.$
(1.10)

The inverse transformation is given by

$\displaystyle x_{i}=\sinh
u_{i}-\left(\frac{1}{2}s_{i}^{2}+\overline{\psi}_{i}\psi_{i}\right)e^{u_{i}},\quad
y_{i}=s_{i}e^{u_{i}},\quad\xi_{i}=\overline{\psi}_{i}e^{u_{i}},\quad\eta_{i}=\psi_{i}e^{u_{i}},$
(1.11) $\displaystyle
z_{i}=\sqrt{1+x^{2}_{i}+y^{2}_{i}+2\xi_{i}\eta_{i}}=\cosh
u_{i}+\left(\frac{1}{2}s_{i}^{2}+\overline{\psi}_{i}\psi_{i}\right)e^{u_{i}}.$
(1.12)

In these coordinates, the even components $(u_{\Lambda},s_{\Lambda})$
restricted to ${\mathbb{R}}^{\Lambda}\times{\mathbb{R}}^{\Lambda}$ have the
following probabilistic interpretation: there is a unique probability measure
$P^{\Lambda}=P_{W,h}^{\Lambda}$ on $({\mathbb{R}}^{\Lambda})^{2}$ such that
for any smooth bounded test function
$f:({\mathbb{R}}^{\Lambda})^{2}\to{\mathbb{R}}$ one has

$\displaystyle\int_{({\mathbb{R}}^{\Lambda})^{2}}f\,dP_{W,h}^{\Lambda}=E_{W,h}^{\Lambda}[f(u_{\Lambda}(v_{\Lambda}),s_{\Lambda}(v_{\Lambda}))].$
(1.13)

Using the abbreviations

$\displaystyle
B_{ij}:=\cosh(u_{i}-u_{j})+\frac{1}{2}(s_{i}-s_{j})^{2}e^{u_{i}+u_{j}},\quad
B_{j}:=\cosh u_{j}+\frac{1}{2}s_{j}^{2}e^{u_{j}},$ (1.14) $\displaystyle
M_{ij}:=1_{\\{i=j\\}}\Big{(}h_{i}e^{u_{i}}+\sum_{k\in\Lambda}W_{ik}e^{u_{i}+u_{k}}\Big{)}-1_{\\{i\neq
j\\}}W_{ij}e^{u_{i}+u_{j}}$ (1.15)

for $i,j\in\Lambda$, the corresponding probability density equals

$\displaystyle
P_{W,h}^{\Lambda}(du_{\Lambda}\,ds_{\Lambda})=e^{-\frac{1}{2}\sum_{i,j\in\Lambda}W_{ij}(B_{ij}-1)}e^{-\sum_{i\in\Lambda}h_{i}(B_{i}-1)}\det(M_{ij})_{i,j\in\Lambda}\prod_{i\in\Lambda}\frac{e^{-u_{i}}\,du_{i}\,ds_{i}}{2\pi},$
(1.16)

where $du_{i}\,ds_{i}$ is the Lebesgue measure on ${\mathbb{R}}^{2}$, cf.
[DSZ10]. In the rest of the paper we mostly work with the $v$-variables.

##  2 Hierarchical $H^{2|2}$-model and results

###  2.1 Definition and tightness

We now consider the special case that $\Lambda$ consists of $2^{N}$ vertices
with some given $N\in{\mathbb{N}}_{0}$, where every vertex $i$ interacts with
every other vertex $j$ in a hierarchical manner. The hierarchical structure is
best visualized by the binary tree with set of leaves equal to the vertex set
$\Lambda$. More precisely, for any $n\in{\mathbb{N}}_{0}$, let
$\Lambda_{n}=\\{0,1\\}^{n}$ be the set of binary words of length $n$. In
particular, $\Lambda_{0}=\\{\emptyset\\}$ and $\Lambda=\Lambda_{N}$. The
reader may visualize the words as binary numbers $0,\ldots,2^{N}-1$ with
reversed order of digits. Let
${\mathcal{T}}^{N}:=\bigcup_{n=0}^{N}\Lambda_{n}$ be the vertex set of a
binary tree of height $N$ with root $\emptyset$. For $j\in\Lambda_{n}$, let
$|j|=n$ be the length of the word $j$, which is the distance of $j$ from the
root. The elements of $\Lambda_{N}$ are the leaves of the tree. An
illustration of the tree ${\mathcal{T}}^{3}$ is given in Figure 1.

Figure 1: The tree
${\mathcal{T}}^{3}=\Lambda_{3}\cup\Lambda_{2}\cup\Lambda_{1}\cup\Lambda_{0}$.

For $j=(j_{0},\ldots,j_{|j|-1})\in{\mathcal{T}}^{N}\setminus\\{\emptyset\\}$,
let ${\tt cut}(j):=(j_{1},\ldots,j_{|j|-1})$ denote the vertex directly below
$j$. The _hierarchical distance_ between $i,j\in\Lambda_{N}$ is defined as

$\displaystyle d(i,j):=\min\\{l\in{\mathbb{N}}_{0}:{\tt cut}^{l}(i)={\tt
cut}^{l}(j)\\}.$ (2.1)

Here, ${\tt cut}^{l}$ denotes the $l$-fold iteration of ${\tt cut}$ and ${\tt
cut}^{0}$ is the identity map. The hierarchical distance is the distance to
the first point in ${\mathcal{T}}^{N}$ where the shortest paths from $i$ and
$j$ to the root meet. It is also half the graph distance between $i$ and $j$
in the tree graph corresponding to ${\mathcal{T}}^{N}$. The _minimum_ of
$i,j\in\Lambda_{N}$ is defined by

$\displaystyle i\wedge j:={\tt cut}^{d(i,j)}(i)={\tt cut}^{d(i,j)}(j).$ (2.2)

More generally, for $i,j\in{\mathcal{T}}^{N}$, we define $i\wedge j$ to be the
least common ancestor of $i$ and $j$, which equals ${\tt cut}^{m}(i)$ with the
smallest $m\in{\mathbb{N}}_{0}$ such that ${\tt cut}^{m}(i)={\tt
cut}^{m^{\prime}}(j)$ for some $m^{\prime}\in{\mathbb{N}}_{0}$.

Take a weight function $w:{\mathbb{N}}_{0}\to[0,\infty)$. It encodes the
weight function $W:{\mathcal{E}}_{\Lambda_{N}}\to[0,\infty)$ on the set of
edges ${\mathcal{E}}_{\Lambda_{N}}$ as follows: for $i,j\in\Lambda_{N}$,

$\displaystyle W_{ij}:=w(d(i,j)).$ (2.3)

Thus, it depends only on the hierarchical distance of $i$ and $j$. In this
paper we study the $H^{2|2}$-model on $\Lambda=\Lambda_{N}$ with hierarchical
weight function $W$ given by (2.3) and a pinning function
$h=(h_{i})_{i\in\Lambda_{N}}$. According to formula (1.7), the action is given
by

$\displaystyle S_{W,h}^{\Lambda_{N}}(v_{\Lambda_{N}})=$
$\displaystyle\sum_{i,j\in\Lambda_{N}}w(d(i,j))(1+\left\langle{v_{i},v_{j}}\right\rangle)+\sum_{i\in\Lambda_{N}}h_{i}(1+\left\langle{v_{i},o}\right\rangle).$
(2.4)

Important particular cases are constant pinning $h=\epsilon
1=(\epsilon)_{i\in\Lambda_{N}}$ and pinning at one point
${i_{0}}\in\Lambda_{N}$, i.e. $h=\epsilon\delta_{i_{0}}=(\epsilon
1_{\\{i={i_{0}}\\}})_{i\in\Lambda_{N}}$, with some pinning strength
$\epsilon>0$. We show that for this particular choice of the pinning the
horospherical coordinates $(u_{i})_{i\in\Lambda_{N}}$ are tight with respect
to $P^{\Lambda_{N}}_{W,h}$, uniformly in $N$ and the pinning strength
$\epsilon>0$ if the weight function $w$ does not decrease too fast.

######  Theorem 2.1 (Tightness)

Let $w:{\mathbb{N}}_{0}\to(0,\infty)$ and
$W:{\mathcal{E}}_{\Lambda_{N}}\to(0,\infty)$ be weight functions related by
(2.3) such that the numbers $\beta_{l}:=2^{2l+1}w(l+2)$,
$l\in{\mathbb{N}}_{0}$, fulfill

$\displaystyle\sum_{l=0}^{\infty}\sqrt{\frac{\log\max\\{\beta_{l},e\\}}{\beta_{l}}}<\infty.$
(2.5)

Then one has

$\displaystyle\lim_{M\to\infty}\sup_{N\in{\mathbb{N}}}\sup_{h}\sup_{p,q\in\Lambda_{N}}P^{\Lambda_{N}}_{W,h}(|u_{p}-u_{q}|\geq
M)=0,$ (2.6)

where the supremum over $h$ is taken over all uniform pinning functions
$h=\epsilon 1$ and all pinning functions at one point
$h=\epsilon\delta_{i_{0}}$ with $\epsilon>0$ and ${i_{0}}\in\Lambda_{N}$.

A key tool for this result is a representation of expectations in terms of an
effective reduced model.

###  2.2 Effective model

It turns out that the value
$E^{\Lambda_{N}}_{W,h}[e^{\sum_{j\in\Lambda}\left\langle{a_{j},v_{j}}\right\rangle}]$
of the superintegration form agrees, for any suitable $a_{j}$ and $h,$ with
the corresponding value of the superintegration form for an $H^{2|2}$-model
with a smaller vertex set, rescaled weights and rescaled pinning. While the
original model has $|\Lambda_{N}|=2^{N}$ vertices, the effective model used in
the proof of Theorem 2.1 has only $O(N)$ vertices, resulting in an exponential
reduction of complexity; see Remark 4.2 for more details. To define this
effective model we need to introduce some notation.

#### Block spin variables.

For $i,j\in{\mathcal{T}}^{N}$, we write $i\preceq j$ if $i={\tt cut}^{l}(j)$
for some $l\in{\mathbb{N}}_{0}$, which means that $j$ is an extension of $i$.
Let $\mathcal{B}_{i}:=\\{j\in\Lambda_{N}:i\preceq j\\}$ denote the set of
leaves above $i$ in $\Lambda_{N}$. In particular,
$\mathcal{B}_{\emptyset}=\Lambda_{N}$. For any vertex $j\in{\mathcal{T}}^{N}$,
let

$\displaystyle\ell(j):=N-|j|$ (2.7)

denote its level. All vertices in $\Lambda_{N}$ are at level 0 and the root
$\emptyset$ is at level $N$; see Figure 1 for an illustration. Note that for
$i\in\Lambda_{n}$, one has $|\mathcal{B}_{i}|=2^{N-n}=2^{\ell(i)}$ and for
$i,j\in\Lambda_{N}$ we have $\ell(i\wedge j)=d(i,j)$. Starting with the
variables $v_{j}$ with $j\in\Lambda_{N}$ at level $0$, we attach _block spin
variables_ to all vertices of the binary tree ${\mathcal{T}}^{N}$ as follows.
For $j\in{\mathcal{T}}^{N}$, let

$\displaystyle
v_{j}=(x_{j},y_{j},z_{j},\xi_{j},\eta_{j}):=\frac{1}{|\mathcal{B}_{j}|}\sum_{i\in\mathcal{B}_{j}}v_{i}=\frac{1}{2^{\ell(j)}}\sum_{i\in\mathcal{B}_{j}}v_{i}.$
(2.8)

For $j\in\Lambda_{N}$, this redefinition preserves the original meaning of
$v_{j}$.

#### Antichains.

For the effective model, we choose the vertex set
$A\subseteq{\mathcal{T}}^{N}$ to be an antichain, meaning that for all $i,j\in
A$ with $i\neq j$ one has $i\not\preceq j$ and $j\not\preceq i$. An antichain
$A$ is called maximal if for any antichain $A^{\prime}\supseteq A$ one has
$A^{\prime}=A$. Equivalently, $A\subseteq{\mathcal{T}}^{N}$ is a maximal
antichain if and only if for every leaf $i\in\Lambda_{N}$ there is precisely
one $j\in A$ with $j\preceq i$. For a maximal antichain $A$, the set
$A^{\uparrow}:=\\{i\in{\mathcal{T}}^{N}:j\preceq i\text{ for some }j\in A\\}$
consists of all vertices above $A$. Given two maximal antichains $A$ and
$A^{\prime}$, we write $A\preceq A^{\prime}$ if $A^{\prime}\subseteq
A^{\uparrow}$. The notation $A\prec A^{\prime}$ means that $A\preceq
A^{\prime}$ and $A\neq A^{\prime}$.

#### Rescaled weights and pinning.

We extend the weight function $W:{\mathcal{E}}_{\Lambda_{N}}\to[0,\infty)$ to
$W:\\{\\{i,j\\}:i,j\in{\mathcal{T}}^{N}\\}\to[0,\infty)$ by

$\displaystyle W_{ij}:=2^{\ell(i)+\ell(j)}w(\ell(i\wedge j)).$ (2.9)

Note that, for $i,j\in\Lambda_{N}$ this definition coincides with the original
weight function $W_{ij}=w(d(i,j)).$

Assume now $h:\Lambda_{N}\to[0,\infty)$ is a pinning function such that
$h_{j}>0$ for at least one point in each connected component of the graph
$(\Lambda_{N},{\mathcal{E}}_{\Lambda_{N},W})$ (cf. remarks below (1.6)). We
call a maximal antichain $A$ _compatible with $h$_, if for all $j\in A$ the
restriction $h|_{\mathcal{B}_{j}}$ of $h$ to $\mathcal{B}_{j}$ is constant;
see Figure 2 for an illustration.

Given a maximal antichain $A$ compatible with a pinning function $h$, we
define an extended pinning function $H:A^{\uparrow}\to[0,\infty)$ by setting

$\displaystyle H_{j}=\sum_{i\in\mathcal{B}_{j}}h_{i}\text{ for }j\in
A^{\uparrow}.$ (2.10)

Note that all summands in the last sum are equal and hence
$H_{j}=2^{\ell(j)}h_{i}$ for any $i\in\mathcal{B}_{j}$, $j\in A^{\uparrow}$.

For any $\Lambda\subseteq A$ the notation $S^{\Lambda}_{W,H}$,
$E^{\Lambda}_{W,H}$ means the weight function $W$ is restricted to the
underlying edge set ${\mathcal{E}}_{\Lambda}$ and the pinning $H$ is
restricted to the vertex set $\Lambda$.

#### Reduction of the model.

We now compare the hierarchical $H^{2|2}$-model over the set $\Lambda_{N}$ of
leaves of the tree ${\mathcal{T}}^{N}$ with arbitrary pinning $h$ with the
$H^{2|2}$-model over a maximal antichain $A\subseteq{\mathcal{T}}^{N}$
compatible with $h$.

Recall the even and odd Grassmann subspaces $\mathcal{A}_{0}$ and
$\mathcal{A}_{1}$, respectively, and
$H^{3|2}=\mathcal{A}_{0}^{3}\times\mathcal{A}_{1}^{2}$ endowed with the inner
product $\left\langle{\cdot,\cdot}\right\rangle$ defined in equation (1.1). We
extend the comparison relations $<,\leq,>,\geq$ on ${\mathbb{R}}$ to even
elements $a,b\in\mathcal{A}_{0}$ by defining $a<b$ to mean
$\operatorname{body}(a)<\operatorname{body}(b)$, and similarly for $\leq$,
$>$, and $\geq$. We abbreviate

$\displaystyle H^{3|2}_{+}:=$ $\displaystyle\\{v=(x,y,z,\xi,\eta)\in
H^{3|2}:\left\langle{v,v}\right\rangle<0,z>0\\},$ (2.11) $\displaystyle
H^{3|2}_{+0}:=$ $\displaystyle\\{v=(x,y,z,\xi,\eta)\in
H^{3|2}:\left\langle{v,v}\right\rangle\leq 0,z\geq 0\\}.$ (2.12)

For the next theorem and its corollary, consider the extended weight function
$W$ from (2.9), a pinning function $h:\Lambda_{N}\to[0,\infty)$, a maximal
antichain $A\subseteq{\mathcal{T}}^{N}$ being compatible with $h$, and the
extension $H$ of $h$ given in (2.10). In analogy to (1.6), we consider the
graph $(A,{\mathcal{E}}_{A,W})$ with edge set
${\mathcal{E}}_{A,W}:=\\{\\{i,j\\}:i,j\in A,W_{ij}>0\\}$.

Figure 2: The antichain consisting of the big dots is compatible with pinning
functions given by $h_{i}$, $1\leq i\leq 7$.

######  Theorem 2.2 (Reduction to the effective model)

For all $(a_{j})_{j\in A}\in(H^{3|2})^{A}$ such that $H_{j}o+a_{j}\in
H^{3|2}_{+0}$ for all $j\in A$ and $H_{j}o+a_{j}\in H^{3|2}_{+}$ for at least
one $j$ in every connected component of the graph $(A,{\mathcal{E}}_{A,W})$,
one has

$\displaystyle E^{\Lambda_{N}}_{W,h}[e^{\sum_{j\in
A}\left\langle{a_{j},v_{j}}\right\rangle}]=E^{A}_{W,H}[e^{\sum_{j\in
A}\left\langle{a_{j},v_{j}}\right\rangle}].$ (2.13)

Note that in formula (2.13) the variables $v_{j}$ have different meanings on
the left-hand side and on the right-hand side. More precisely, on the left-
hand side, they are understood as block spin variables as in formula (2.8),
while on the right-hand side they are the original variables $v_{j}\in
H^{2|2}$ of the $H^{2|2}$-model with vertex set $A$.

We remark that taking derivatives with respect to the components of the
parameters $a_{j}$ in formula (2.13), one can treat also polynomials in the
$x_{j},y_{j},z_{j},\xi_{j},\eta_{j}$. We expect that this can be extended to
general observables $f((v_{j})_{j\in A})$ rather than only $e^{\sum_{j\in
A}\left\langle{a_{j},v_{j}}\right\rangle}$ by working out a theory for super
Fourier Laplace transforms of super measures in the spirit of [Fre21].
However, for our application, we need only a very special case which is
treated in the following corollary in an elementary way.

Let $\mathcal{L}_{P}((X_{i})_{i\in I})$ denote the joint law of the random
variables $(X_{i})_{i\in I}$ with respect to some probability law $P$. Recall
the assumptions specified right above Theorem 2.2.

######  Corollary 2.3 (Coincidence of distributions)

Let $B:=A\cap\Lambda_{N}$ denote the set of leaves of the maximal antichain
$A$. Then, the joint laws of $u_{B}:=(u_{j})_{j\in B}$ and
$s_{B}:=(s_{j})_{j\in B}$ with respect to $P^{\Lambda_{N}}_{W,h}$ and
$P^{A}_{W,H}$ coincide. More generally, the following joint laws agree

$\displaystyle\mathcal{L}_{P^{\Lambda_{N}}_{W,h}}\Big{(}\Big{(}|\mathcal{B}_{j}|^{-1}\sum_{i\in\mathcal{B}_{j}}e^{u_{i}},|\mathcal{B}_{j}|^{-1}\sum_{i\in\mathcal{B}_{j}}s_{i}e^{u_{i}}\Big{)}_{j\in
A}\Big{)}=\mathcal{L}_{P^{A}_{W,H}}((e^{u_{j}},s_{j}e^{u_{j}})_{j\in A}).$
(2.14)

##  3 Derivation of the effective model

###  3.1 Reduction of certain integrals

The (super-)symmetries of the $H^{2|2}$-model allow to calculate certain
integrals. Lemma 3.2 below provides such a result. It is crucially used in the
proof of Theorem 2.2.

######  Definition 3.1 (Fast decaying superfunctions)

For any finite set $\Lambda$, a function
$f:(H^{3|2})^{\Lambda}\to\mathcal{A}_{0}$, possibly depending on parameters,
is called fast decaying on $(H^{2|2})^{\Lambda}$ if it is a smooth
superfunction and all coefficients of
$({\mathbb{R}}^{2})^{\Lambda}\ni(x_{j},y_{j})_{j\in\Lambda}\mapsto
f((x_{j},y_{j},\sqrt{1+x_{j}^{2}+y_{j}^{2}+2\xi_{j}\eta_{j}},\xi_{j},\eta_{j})_{j\in\Lambda})$
in the sense of formula (A.1) in the appendix are Schwartz functions.

For $a\in H^{3|2}_{+}$, we set
$\|a\|:=\sqrt{-\left\langle{a,a}\right\rangle}$. Note that for $a\in
H^{3|2}_{+0}$ with $\operatorname{body}\left\langle{a,a}\right\rangle=0$ and
non zero Grassmann components the norm cannot be defined. Recall the
abbreviation $\mathcal{D}v_{\Lambda}$ introduced in formula (1.9).

######  Lemma 3.2 (Reduction to single spins)

For any finite set $\Lambda$, $c_{j}\in[0,\infty)$, $j\in\Lambda$, $a\in
H^{3|2}_{+}$, and any function $f:(H^{3|2}_{+})^{\Lambda}\to\mathcal{A}_{0}$
such that $v_{\Lambda}\mapsto
f\left((\left\langle{v_{i},v_{j}}\right\rangle)_{i,j\in\Lambda}\right)e^{\sum_{j\in\Lambda}c_{j}\left\langle{v_{j},a}\right\rangle}$
is fast decaying on $(H^{2|2})^{\Lambda}$, one has

$\displaystyle\int_{(H^{2|2})^{\Lambda}}\mathcal{D}v_{\Lambda}\,f\left((\left\langle{v_{i},v_{j}}\right\rangle)_{i,j\in\Lambda}\right)e^{\sum_{j\in\Lambda}c_{j}\left\langle{v_{j},a}\right\rangle}$
$\displaystyle=$ $\displaystyle
f\left((-1)_{i,j\in\Lambda}\right)e^{-\sum_{j\in\Lambda}c_{j}\|a\|}=\int_{H^{2|2}}\mathcal{D}v\,f\left((\left\langle{v,v}\right\rangle)_{i,j\in\Lambda}\right)e^{\sum_{j\in\Lambda}c_{j}\left\langle{v,a}\right\rangle}.$
(3.1)

This lemma has the following simple but interesting consequence.

######  Corollary 3.3 (Super Laplace transform of a block spin)

Let $\epsilon>0$ and consider the $H^{2|2}$-model on an arbitrary finite set
$\Lambda$ with arbitrary weights $W$ and uniform pinning given by
$h_{j}=|\Lambda|^{-1}\epsilon$ for all $j\in\Lambda$. For all $a\in H^{3|2}$
with $a+\epsilon o\in H^{3|2}_{+}$, the super Laplace transform of
$|\Lambda|^{-1}\sum_{j\in\Lambda}v_{j}$ is given by

$\displaystyle
E^{\Lambda}_{W,h}\left[e^{\left\langle{a,|\Lambda|^{-1}\sum_{j\in\Lambda}v_{j}}\right\rangle}\right]=e^{\epsilon-\|a+\epsilon
o\|}=e^{\epsilon-\sqrt{\epsilon^{2}-2\epsilon\left\langle{a,o}\right\rangle-\left\langle{a,a}\right\rangle}}.$
(3.2)

In particular, the mean
$|\Lambda|^{-1}\sum_{j\in\Lambda}(x_{j}+z_{j})=|\Lambda|^{-1}\sum_{j\in\Lambda}e^{u_{j}}$
has the probability density

$\displaystyle f_{1,\epsilon}(t)=\sqrt{\frac{\epsilon}{2\pi
t^{3}}}\exp\left(-\frac{\epsilon(t-1)^{2}}{2t}\right),\quad t>0,$ (3.3)

i.e. it is an inverse Gaussian distribution with parameters $1$ and $\epsilon$
with respect to $P^{\Lambda}_{W,h}$.

We now give the proofs of the two results.

Proof of Lemma 3.2. It suffices to prove the first equality in (3.1), because
the second one is just the special case of $\Lambda$ being a singleton. Let
$F:H^{3|2}_{+}\times(H^{3|2}_{+})^{\Lambda}\to\mathcal{A}_{0}$ be defined by

$\displaystyle F(a,v_{\Lambda}):=$ $\displaystyle
f\left((\left\langle{v_{i},v_{j}}\right\rangle)_{i,j\in\Lambda}\right)e^{\sum_{j\in\Lambda}c_{j}\left\langle{v_{j},a}\right\rangle}.$
(3.4)

Recall $o$ from equation (1.5).

The proof uses the supersymmetry operators $L_{ij}$ (cf. Def. A.1 in the
appendix), and is organized in four steps, according to the form of the vector
$a.$ Precisely $a=(0,0,z_{a},0,0)$ in case 1, $a=(x_{a},0,z_{a},0,0)$ in case
2, $a=(x_{a},y_{a},z_{a},0,0)$ in case 3 and finally $a$ has the most general
form $a=(x_{a},y_{a},z_{a},\xi_{a},\eta_{a})$ in case 4.

_Case 1._ Let $a=\alpha o,$ where the parameter $\alpha$ is an even element of
the Grassmann algebra with $\alpha>0,$ and must not depend on the integration
variables $v_{j}$. In this case, by formula (A.21) (from Fact A.2 in the
appendix), the integrand $F$ is annihilated by the odd supersymmetry operator
$Q:=\sum_{j\in\Lambda}Q^{v_{j}}$, where $Q^{v_{j}}$ is defined in (A.17). The
hypothesis that $F(a,\cdot)$ is fast decaying allows us to apply the
localization result from [DSZ10][Appendix C] (in particular Proposition 2) as
follows

$\displaystyle\int_{(H^{2|2})^{\Lambda}}\mathcal{D}v_{\Lambda}\,f\left((\left\langle{v_{i},v_{j}}\right\rangle)_{i,j\in\Lambda}\right)e^{\sum_{j\in\Lambda}c_{j}\left\langle{v_{j},a}\right\rangle}$
$\displaystyle\qquad\qquad=f\left((\left\langle{o,o}\right\rangle)_{i,j\in\Lambda}\right)e^{\sum_{j\in\Lambda}c_{j}\left\langle{o,a}\right\rangle}=f\left((-1)_{i,j\in\Lambda}\right)e^{-\sum_{j\in\Lambda}c_{j}\|a\|}.$
(3.5)

_Case 2._ Let $a=(x_{a},0,z_{a},0,0)$ with $x_{a},z_{a}$ even elements,
$z_{a}>0$, $\left\langle{a,a}\right\rangle<0$. We define
$\varphi:=\operatorname{artanh}\frac{x_{a}}{z_{a}}$, and set, for $0\leq t\leq
1$, $t\in{\mathbb{R}}$,

$\displaystyle a(t)=(x(t),0,z(t),0,0):=\sqrt{z_{a}^{2}-x_{a}^{2}}\
\big{(}\sinh(t\varphi),0,\cosh(t\varphi),0,0\big{)},$ (3.6)

where $\sqrt{z_{a}^{2}-x_{a}^{2}}=\sqrt{-\left\langle{a,a}\right\rangle}$ is
well defined because $\left\langle{a,a}\right\rangle<0$. Note that
$\left\langle{a(t),a(t)}\right\rangle=\left\langle{a,a}\right\rangle$ for all
$t$. Moreover,

$\displaystyle
a(0)=\sqrt{z_{a}^{2}-x_{a}^{2}}(0,0,1,0,0)=\sqrt{z_{a}^{2}-x_{a}^{2}}\
o\quad\text{ and }\quad a(1)=a.$ (3.7)

By direct computation one has

$\displaystyle\frac{d}{dt}F(a(t),v_{\Lambda})=-\varphi
L_{13}^{a(t)}F(a(t),v_{\Lambda}),$ (3.8)

where $L_{13}^{a(t)}$ acts on the first component of $F$, which is then
substituted by $a(t)$. By formula (A.19) in Fact A.2, one has

$\displaystyle\Big{(}L_{13}^{a(t)}+\sum_{j\in\Lambda}L_{13}^{v_{j}}\Big{)}F(a(t),v_{\Lambda})=0$
(3.9)

because the variables appear only inside the inner product. We conclude

$\displaystyle\frac{d}{dt}\int_{(H^{2|2})^{\Lambda}}\mathcal{D}v_{\Lambda}\,F(a(t),v_{\Lambda})=\int_{(H^{2|2})^{\Lambda}}\mathcal{D}v_{\Lambda}\,\frac{d}{dt}F(a(t),v_{\Lambda})$
$\displaystyle=-\int_{(H^{2|2})^{\Lambda}}\mathcal{D}v_{\Lambda}\,\varphi
L_{13}^{a(t)}F(a(t),v_{\Lambda})=\varphi\int_{(H^{2|2})^{\Lambda}}\mathcal{D}v_{\Lambda}\,\sum_{j\in\Lambda}L_{13}^{v_{j}}F(a(t),v_{\Lambda})=0,$
(3.10)

where in the last step we have used the Ward identity (A.20). Consequently,
the function

$\displaystyle[0,1]\ni
t\mapsto\int_{(H^{2|2})^{\Lambda}}\mathcal{D}v_{\Lambda}\,F(a(t),v_{\Lambda})$
(3.11)

is constant and in particular, its values at $t=0$ and $t=1$ agree. Recall the
expressions (3.7) for $a(0)$ and $a(1)$. Using that Case 1 is applicable to
$a(0)$, we conclude

$\displaystyle\int_{(H^{2|2})^{\Lambda}}\mathcal{D}v_{\Lambda}\,F(a,v_{\Lambda})=\int_{(H^{2|2})^{\Lambda}}\mathcal{D}v_{\Lambda}\,F(a(1),v_{\Lambda})$
(3.12)
$\displaystyle\quad=\int_{(H^{2|2})^{\Lambda}}\mathcal{D}v_{\Lambda}\,F(a(0),v_{\Lambda})=F\left((-1)_{i,j\in\Lambda}\right)e^{-\sum_{j\in\Lambda}c_{j}\|a(0)\|}=F\left((-1)_{i,j\in\Lambda}\right)e^{-\sum_{j\in\Lambda}c_{j}\|a\|}.$

_Case 3:_ Let $a=(x_{a},y_{a},z_{a},0,0)$ with $x_{a},y_{a},z_{a}$ even
elements, $z_{a}>0$, $\left\langle{a,a}\right\rangle<0$. Set
$\varphi=\operatorname{artanh}\frac{y_{a}}{z_{a}}$ and, for $0\leq t\leq 1$,
$t\in{\mathbb{R}}$,

$\displaystyle a(t)=(x_{a},y(t),z(t),0,0)$ (3.13)
$\displaystyle\text{with}\quad
y(t)=\sqrt{z_{a}^{2}-y_{a}^{2}}\sinh(t\varphi),\quad
z(t)=\sqrt{z_{a}^{2}-y_{a}^{2}}\cosh(t\varphi),$ (3.14)

where $\sqrt{z_{a}^{2}-y_{a}^{2}}$ is well defined because
$z_{a}^{2}-y_{a}^{2}\geq-\left\langle{a,a}\right\rangle>0$. Note that
$\left\langle{a(t),a(t)}\right\rangle=\left\langle{a,a}\right\rangle$ for all
$t$. Moreover,

$\displaystyle a(0)=(x_{a},0,\sqrt{z_{a}^{2}-y_{a}^{2}},0,0)\quad\text{ and
}\quad a(1)=a.$ (3.15)

As in Case 2, by direct computation one has

$\displaystyle\frac{d}{dt}F(a(t),v_{\Lambda})=-\varphi
L_{23}^{a(t)}F(a(t),v_{\Lambda})$ (3.16)

and again formula (A.19) in Fact A.2 implies

$\displaystyle\Big{(}L_{23}^{a(t)}+\sum_{j\in\Lambda}L_{23}^{v_{j}}\Big{)}F(a(t),v_{\Lambda})=0.$
(3.17)

Using the same argument as in (3.10), with $L_{13}$ replaced by $L_{23}$, we
conclude

$\displaystyle\frac{d}{dt}\int_{(H^{2|2})^{\Lambda}}\mathcal{D}v_{\Lambda}\,F(a(t),v_{\Lambda})=0.$
(3.18)

Using that Case 2 is applicable to $a(0)$, the claim follows.
_Case 4:_ Finally, we consider the most general case
$a=(x_{a},y_{a},z_{a},\xi_{a},\eta_{a})$. As in the previous cases, we define
for $0\leq t\leq 1$,

$\displaystyle a(t)=(x_{a},y_{a},z(t),\xi(t),\eta(t))\quad\text{with}$ (3.19)
$\displaystyle
z(t)=z_{a}-\frac{1-t^{2}}{z_{a}}\xi_{a}\eta_{a}=\sqrt{z_{a}^{2}-(1-t^{2})2\xi_{a}\eta_{a}},\quad\xi(t)=t\xi_{a},\quad\eta(t)=t\eta_{a}.$
(3.20)

Note that
$\left\langle{a(t),a(t)}\right\rangle=\left\langle{a,a}\right\rangle$ for all
$t$. Moreover,

$\displaystyle
a(0)=(x_{a},y_{a},z_{a}-\tfrac{1}{z_{a}}\xi_{a}\eta_{a},0,0)=(x_{a},y_{a},\sqrt{z_{a}^{2}-2\xi_{a}\eta_{a}},0,0)\quad\text{
and }\quad a(1)=a.$ (3.21)

Below we denote by $\partial_{z(t)}F(a(t),v_{\Lambda})$ the derivative with
respect to the $z$-component of $a(t)$, which is then substituted by $z(t)$.
The same notational convention holds for $\partial_{\xi(t)}$ and
$\partial_{\eta(t)}$. Analogue to Cases 2 and 3, using
$\xi_{a}z(t)=\xi_{a}z_{a}$ and $\eta_{a}z(t)=\eta_{a}z_{a}$, we get

$\displaystyle\frac{d}{dt}F(a(t),v_{\Lambda})=\left[z^{\prime}(t)\partial_{z(t)}+\xi^{\prime}(t)\partial_{\xi(t)}+\eta^{\prime}(t)\partial_{\eta(t)}\right]F(a(t),v_{\Lambda})$
$\displaystyle=$
$\displaystyle\left[\frac{2t}{z_{a}}\xi_{a}\eta_{a}\partial_{z(t)}+\xi_{a}\partial_{\xi(t)}+\eta_{a}\partial_{\eta(t)}\right]F(a(t),v_{\Lambda})$
$\displaystyle=$
$\displaystyle\frac{1}{z_{a}}\big{[}\xi_{a}\big{(}z(t)\partial_{\xi(t)}+\eta(t)\partial_{z(t)}\big{)}+\eta_{a}\big{(}z(t)\partial_{\eta(t)}-\xi(t)\partial_{z(t)}\big{)}\big{]}F(a(t),v_{\Lambda})$
$\displaystyle=$
$\displaystyle-\frac{1}{z_{a}}[\xi_{a}L_{34}^{a(t)}+\eta_{a}L_{35}^{a(t)}]F(a(t),v_{\Lambda}).$
(3.22)

Note that $\tfrac{\xi_{a}}{z_{a}}$ and $\tfrac{\eta_{a}}{z_{a}}$ are the
analogue of $\varphi$ in cases 2 and 3. Formula (A.19) implies

$\displaystyle\Big{(}L_{ik}^{a(t)}+\sum_{j\in\Lambda}L_{ik}^{v_{j}}\Big{)}F(a(t),v_{\Lambda})=0$
(3.23)

for $ik=34,35$. Using again the Ward identity from Fact A.2 and the fact that
Case 3 is applicable to $a(0)$ we conclude.

As a first application of Lemma 3.2, we determine the super Laplace transform
of $|\Lambda|^{-1}\sum_{j\in\Lambda}v_{j}$.

Proof of Corollary 3.3. Using first the definition (1.8) of the
superintegration form and then Lemma 3.2 with the function
$f\left((\left\langle{v_{i},v_{j}}\right\rangle)_{i,j\in\Lambda}\right)=e^{\sum_{i,j\in\Lambda}W_{ij}(1+\left\langle{v_{i},v_{j}}\right\rangle)+\epsilon}$,
$c_{j}=|\Lambda|^{-1}$ for all $j$, and $a$ replaced by $a+\epsilon o\in
H^{3|2}_{+}$, we obtain

$\displaystyle
E^{\Lambda}_{W,h}\left[e^{\left\langle{a,|\Lambda|^{-1}\sum_{j\in\Lambda}v_{j}}\right\rangle}\right]=$
$\displaystyle\int_{(H^{2|2})^{\Lambda}}\mathcal{D}v_{\Lambda}e^{\sum_{i,j\in\Lambda}W_{ij}(1+\left\langle{v_{i},v_{j}}\right\rangle)+|\Lambda|^{-1}\epsilon\sum_{i\in\Lambda}(1+\left\langle{v_{i},o}\right\rangle)}e^{\left\langle{a,|\Lambda|^{-1}\sum_{j\in\Lambda}v_{j}}\right\rangle}$
$\displaystyle=$ $\displaystyle e^{\epsilon-\|a+\epsilon
o\|}=\exp(\epsilon-\sqrt{\epsilon^{2}-2\epsilon\left\langle{a,o}\right\rangle-\left\langle{a,a}\right\rangle}).$
(3.24)

We consider $a=(b,0,-b,0,0)$ with $b<0$. Then, we find
$\left\langle{a,a}\right\rangle=0$, $\left\langle{a,o}\right\rangle=b$,
$a+\epsilon o\in H^{3|2}_{+}$ and
$be^{u_{j}}=b(x_{j}+z_{j})=\left\langle{a,v_{j}}\right\rangle$. Consequently,
the Laplace transform of
$|\Lambda|^{-1}\sum_{j\in\Lambda}e^{u_{j}}=|\Lambda|^{-1}\sum_{j\in\Lambda}(x_{j}+z_{j})$
is given by

$\displaystyle
E^{\Lambda}_{W,h}\left[e^{b|\Lambda|^{-1}\sum_{j\in\Lambda}e^{u_{j}}}\right]=E^{\Lambda}_{W,h}\left[e^{\left\langle{a,|\Lambda|^{-1}\sum_{j\in\Lambda}v_{j}}\right\rangle}\right]=e^{\epsilon-\sqrt{\epsilon^{2}-2\epsilon
b}}.$ (3.25)

This is the Laplace transform of an inverse Gaussian distribution with
parameters $1$ and $\epsilon$ and the claim follows.

###  3.2 Reduction to the effective model

Using the reduction of integrals given in Lemma 3.2 and a representation of
the action of the hierarchical $H^{2|2}$-model, we prove Theorem 2.2.

Proof of Theorem 2.2. The conditions on $H_{j}o+a_{j}$ guarantee that both
integrals in (2.13) are well-defined and finite. Let $\mathcal{A}_{h}$ denote
the set of maximal antichains which are compatible with $h$ and recall the
partial order $\prec$ for antichains defined in Section 2.2. Since there do
not exist infinite increasing sequences $A_{1}\prec A_{2}\prec
A_{3}\prec\cdots$ in $\mathcal{A}_{h}$, we can prove the claim by induction
over $\mathcal{A}_{h}$ with respect to the reversed partial order $\succ$ of
$\prec$.

Let $A\in\mathcal{A}_{h}$. As induction hypothesis, assume that the claim
holds for all maximal antichains $A^{\prime}\in\mathcal{A}_{h}$ with $A\prec
A^{\prime}$. If $A=\Lambda_{N}$, there is nothing to prove. Otherwise, we find
$i\in A\setminus\Lambda_{N}$ and set
$A^{\prime}=(A\setminus\\{i\\})\cup\mathcal{B}_{i}$. Note that $A^{\prime}$ is
a maximal antichain compatible with $h$ and fulfilling $A\prec A^{\prime}.$
Set $a_{j}^{\prime}=a_{j}$ for $j\in A\setminus\\{i\\}$ and
$a_{j}^{\prime}=|\mathcal{B}_{i}|^{-1}a_{i}$ for $j\in\mathcal{B}_{i}$. Then,
$\sum_{j\in\mathcal{B}_{i}}\left\langle{a_{j}^{\prime},v_{j}}\right\rangle=\left\langle{a_{i},v_{i}}\right\rangle$
by the definition (2.8) of the block spin variable $v_{i}$, and hence
$\sum_{j\in A}\left\langle{a_{j},v_{j}}\right\rangle=\sum_{j\in
A^{\prime}}\left\langle{a_{j}^{\prime},v_{j}}\right\rangle$. Moreover, for
$j\in B_{i},$ the equality $H_{j}=h_{j}$ and $a_{i}+H_{i}o\in
H^{3|2}_{+0},H^{3|2}_{+}$ imply $a^{\prime}_{j}+h_{j}o\in
H^{3|2}_{+0},H^{3|2}_{+},$ respectively. Consequently, the induction
hypothesis is applicable to $A^{\prime}$ and the $a_{j}^{\prime}$. This yields

$\displaystyle E^{\Lambda_{N}}_{W,h}\left[e^{\sum_{j\in
A}\left\langle{a_{j},v_{j}}\right\rangle}\right]=E^{\Lambda_{N}}_{W,h}\left[e^{\sum_{j\in
A^{\prime}}\left\langle{a_{j}^{\prime},v_{j}}\right\rangle}\right]=E^{A^{\prime}}_{W,H}\left[e^{\sum_{j\in
A^{\prime}}\left\langle{a_{j}^{\prime},v_{j}}\right\rangle}\right].$ (3.26)

By the definition (1.7) of the action $S_{W,H}^{A^{\prime}}(v_{A^{\prime}})$
with vertex set $A^{\prime}$, weights $W_{ij}$ and pinning $H_{j}$ given in
(2.9) and (2.10), respectively, we obtain

$\displaystyle S_{W,H}^{A^{\prime}}(v_{A^{\prime}})=\sum_{i,j\in
A^{\prime}}W_{ij}(1+\left\langle{v_{i},v_{j}}\right\rangle)+\sum_{j\in
A^{\prime}}H_{j}(1+\left\langle{v_{j},o}\right\rangle)$ (3.27)
$\displaystyle=$ $\displaystyle
S_{W,H}^{A\setminus\\{i\\}}(v_{A\setminus\\{i\\}})+\sum_{j,k\in\mathcal{B}_{i}}W_{jk}(1+\left\langle{v_{j},v_{k}}\right\rangle)+2\sum_{\begin{subarray}{c}j\in
A\setminus\\{i\\}\\\
k\in\mathcal{B}_{i}\end{subarray}}W_{jk}(1+\left\langle{v_{j},v_{k}}\right\rangle)+\sum_{k\in\mathcal{B}_{i}}H_{k}(1+\left\langle{v_{k},o}\right\rangle).$

Note that for $j\in A\setminus\\{i\\}$ and $k\in\mathcal{B}_{i}$, one has
$j\wedge k=j\wedge i$ because $j\in A$ and $A$ is an antichain. It follows
that $W_{jk}=2^{\ell(j)+\ell(k)}w(\ell(j\wedge k))=2^{\ell(j)}w(\ell(j\wedge
i))=2^{-\ell(i)}W_{ij},$ which is independent of $k.$ Furthermore, since $A$
is compatible with $h$, one has the same value $H_{k}=2^{-\ell(i)}H_{i}$ for
all $k\in\mathcal{B}_{i}.$ Hence, the last two summands in (3.27) can be
written as follows

$2\sum_{\begin{subarray}{c}j\in A\setminus\\{i\\}\\\
k\in\mathcal{B}_{i}\end{subarray}}W_{jk}(1+\left\langle{v_{j},v_{k}}\right\rangle)+\sum_{k\in\mathcal{B}_{i}}H_{k}(1+\left\langle{v_{k},o}\right\rangle)=C_{i}+\sum_{k\in\mathcal{B}_{i}}2^{-\ell(i)}\left\langle{v_{k},V_{i}}\right\rangle,$
(3.28)

where

$\displaystyle C_{i}=H_{i}+\sum_{j\in A\setminus\\{i\\}}2W_{ij},\qquad
V_{i}=H_{i}o+\sum_{j\in A\setminus\\{i\\}}2W_{ij}v_{j},$ (3.29)

and we used, for each $k\in\mathcal{B}_{i},$

$\displaystyle\sum_{j\in
A\setminus\\{i\\}}2W_{jk}v_{j}+H_{k}o=2^{-\ell(i)}\Big{(}H_{i}o+\sum_{j\in
A\setminus\\{i\\}}2W_{ij}v_{j}\Big{)}=2^{-\ell(i)}V_{i}.$ (3.30)

Substituting this in (3.27) yields

$\displaystyle S_{W,H}^{A^{\prime}}(v_{A^{\prime}})=$ $\displaystyle
S_{W,H}^{A\setminus\\{i\\}}(v_{A\setminus\\{i\\}})+C_{i}+\sum_{j,k\in\mathcal{B}_{i}}W_{jk}(1+\left\langle{v_{j},v_{k}}\right\rangle)+\sum_{k\in\mathcal{B}_{i}}2^{-\ell(i)}\left\langle{v_{k},V_{i}}\right\rangle.$
(3.31)

Using first the definition of the superintegration form (1.8) and then this
representation and the fact $a_{j}^{\prime}=2^{-\ell(i)}a_{i}$ for
$j\in\mathcal{B}_{i}$, we obtain

$\displaystyle E^{A^{\prime}}_{W,H}\left[e^{\sum_{j\in
A^{\prime}}\left\langle{a_{j}^{\prime},v_{j}}\right\rangle}\right]=\int\limits_{(H^{2|2})^{A^{\prime}}}\mathcal{D}v_{A^{\prime}}\,e^{S^{A^{\prime}}_{W,H}(v_{A^{\prime}})}e^{\sum_{j\in
A^{\prime}}\left\langle{a_{j}^{\prime},v_{j}}\right\rangle}$
$\displaystyle\qquad=\int\limits_{(H^{2|2})^{A\setminus\\{i\\}}}\mathcal{D}v_{A\setminus\\{i\\}}\,e^{S_{W,H}^{A\setminus\\{i\\}}(v_{A\setminus\\{i\\}})}e^{\sum_{j\in
A\setminus\\{i\\}}\left\langle{a_{j}^{\prime},v_{j}}\right\rangle}e^{C_{i}}$
$\displaystyle\qquad\qquad\cdot\int\limits_{(H^{2|2})^{\mathcal{B}_{i}}}\mathcal{D}v_{\mathcal{B}_{i}}\,e^{\sum_{j,k\in\mathcal{B}_{i}}W_{jk}(1+\left\langle{v_{j},v_{k}}\right\rangle)}e^{\sum_{k\in\mathcal{B}_{i}}2^{-\ell(i)}\langle
v_{k},V_{i}+a_{i}\rangle}.$ (3.32)

We apply Lemma 3.2 to the inner integral with $\Lambda=\mathcal{B}_{i}$, the
function
$f((\left\langle{v_{j},v_{k}}\right\rangle)_{j,k\in\mathcal{B}_{i}})=e^{\sum_{j,k\in\mathcal{B}_{i}}W_{jk}(1+\left\langle{v_{j},v_{k}}\right\rangle)}$,
$c_{j}=2^{-\ell(i)}$, and $a=V_{i}+a_{i}$. To see that $a\in H^{3|2}_{+}$ we
distinguish two cases. In the case $W_{ij}=0$ for all $j\in
A\setminus\\{i\\}$, all these points are in different connected components;
hence the assumption on the pinning guarantees $a=H_{i}o+a_{i}\in
H^{3|2}_{+}$. Otherwise, we have $\sum_{j\in A\setminus\\{i\\}}W_{ij}v_{j}\in
H^{3|2}_{+}$, which implies $a\in H^{3|2}_{+}$ because of $H_{i}o+a_{i}\in
H^{3|2}_{+0}$. Lemma 3.2 allows us to replace the multidimensional integration
with respect to $\mathcal{D}v_{\mathcal{B}_{i}}$ by a single integral with
respect to $\mathcal{D}v_{i}$ over $H^{2|2}$. We obtain

$\displaystyle E^{A^{\prime}}_{W,H}\left[e^{\sum_{j\in
A^{\prime}}\left\langle{a_{j}^{\prime},v_{j}}\right\rangle}\right]$ (3.33)
$\displaystyle=\int\limits_{(H^{2|2})^{A\setminus\\{i\\}}}\mathcal{D}v_{A\setminus\\{i\\}}\,e^{S_{W,H}^{A\setminus\\{i\\}}(v_{A\setminus\\{i\\}})}e^{\sum_{j\in
A\setminus\\{i\\}}\left\langle{a_{j},v_{j}}\right\rangle}e^{C_{i}}\int\limits_{H^{2|2}}\mathcal{D}v_{i}e^{\sum_{j,k\in\mathcal{B}_{i}}W_{jk}(1+\left\langle{v_{i},v_{i}}\right\rangle)}e^{\langle
v_{i},V_{i}+a_{i}\rangle}$
$\displaystyle=\int\limits_{(H^{2|2})^{A\setminus\\{i\\}}}\mathcal{D}v_{A\setminus\\{i\\}}\,e^{S_{W,H}^{A\setminus\\{i\\}}(v_{A\setminus\\{i\\}})}\int\limits_{H^{2|2}}\mathcal{D}v_{i}\,e^{2\sum_{j\in
A\setminus\\{i\\}}W_{ij}(1+\left\langle{v_{i},v_{j}}\right\rangle)}e^{H_{i}(1+\left\langle{v_{i},o}\right\rangle)}e^{\sum_{j\in
A}\left\langle{a_{j},v_{j}}\right\rangle},$

where we used $\left\langle{v_{i},v_{i}}\right\rangle=-1$ for $v_{i}\in
H^{2|2}$ and the definitions of $C_{i}$ and $V_{i}$. To merge the exponents,
we observe that

$\displaystyle
S_{W,H}^{A}(v_{A})=S_{W,H}^{A\setminus\\{i\\}}(v_{A\setminus\\{i\\}})+2\sum_{j\in
A\setminus\\{i\\}}W_{ij}(1+\left\langle{v_{i},v_{j}}\right\rangle)+H_{i}(1+\left\langle{v_{i},o}\right\rangle).$
(3.34)

Together with (3.26) this concludes the induction step:

$\displaystyle E^{A^{\prime}}_{W,H}\left[e^{\sum_{j\in
A^{\prime}}\left\langle{a_{j}^{\prime},v_{j}}\right\rangle}\right]=$
$\displaystyle\int\limits_{(H^{2|2})^{A}}\mathcal{D}v_{A}\,e^{S_{W,H}^{A}(v_{A})}e^{\sum_{j\in
A}\left\langle{a_{j},v_{j}}\right\rangle}=E^{A}_{W,H}\left[e^{\sum_{j\in
A}\left\langle{a_{j},v_{j}}\right\rangle}\right].$ (3.35)

Using the reduction to the effective model, we can prove the identities in
distribution for certain horospherical coordinates as stated in Corollary 2.3.

Proof of Corollary 2.3. First we prove claim (2.14) under the extra assumption
that $h_{i}>0$ for all $i\in\Lambda_{N}$. It suffices to show that the joint
Laplace transforms of
$(|\mathcal{B}_{j}|^{-1}\sum_{i\in\mathcal{B}_{j}}e^{u_{i}},|\mathcal{B}_{j}|^{-1}\sum_{i\in\mathcal{B}_{j}}s_{i}e^{u_{i}})_{j\in
A}$ and $(e^{u_{j}},s_{j}e^{u_{j}})_{j\in A}$ with respect to
$P^{\Lambda_{N}}_{W,h}$ and $P^{A}_{W,H}$, respectively, agree on a non-empty
open set and are finite there. Thus, we will show the following equality for
all $(c_{j})_{j\in A}\in(0,\infty)^{A}$ and all
$c_{j}^{\prime}\in(-H_{j},H_{j})$, $j\in A$:

$\displaystyle E^{\Lambda_{N}}_{W,h}\left[e^{-\sum_{j\in
A}|\mathcal{B}_{j}|^{-1}\sum_{i\in\mathcal{B}_{j}}(c_{j}e^{u_{i}}+c_{j}^{\prime}s_{i}e^{u_{i}})}\right]=E^{A}_{W,H}\left[e^{-\sum_{j\in
A}(c_{j}e^{u_{j}}+c_{j}^{\prime}s_{j}e^{u_{j}})}\right].$ (3.36)

We rewrite this claim in cartesian coordinates. Using
$-c_{j}e^{u_{i}}-c_{j}^{\prime}s_{i}e^{u_{i}}=-c_{j}(x_{i}+z_{i})-c_{j}^{\prime}y_{i}=\left\langle{v_{i},a_{j}}\right\rangle$
with $a_{j}=(-c_{j},-c_{j}^{\prime},c_{j},0,0)$ for $j\in A$,
$i\in\mathcal{B}_{j}$ on the left-hand side and
$-c_{j}e^{u_{j}}-c_{j}^{\prime}s_{j}e^{u_{j}}=\left\langle{v_{j},a_{j}}\right\rangle$
for $j\in A$ on the right-hand side, it takes the following form:

$\displaystyle E^{\Lambda_{N}}_{W,h}\left[e^{\sum_{j\in
A}\left\langle{v_{j},a_{j}}\right\rangle}\right]=E^{\Lambda_{N}}_{W,h}\left[e^{\sum_{j\in
A}|\mathcal{B}_{j}|^{-1}\sum_{i\in\mathcal{B}_{j}}\left\langle{v_{i},a_{j}}\right\rangle}\right]=E^{A}_{W,H}\left[e^{\sum_{j\in
A}\left\langle{v_{j},a_{j}}\right\rangle}\right].$ (3.37)

Since $\left\langle{H_{j}o+a_{j},H_{j}o+a_{j}}\right\rangle=c_{j}^{\prime
2}-2c_{j}H_{j}-H_{j}^{2}<0$, $H_{j}+c_{j}>0$, and thus $H_{j}o+a_{j}\in
H^{3|2}_{+}$, the claim, including finiteness of the expectations in (3.37),
follows from Theorem 2.2.

Now we drop the extra assumption $h_{i}>0$ for all $i\in\Lambda_{N}$. Let
${\mathcal{N}}:=\\{i\in\Lambda_{N}:h_{i}>0\\}$,

$\displaystyle\mathcal{H}:=\\{(h_{i}^{\prime})_{i\in\Lambda_{N}}:h_{i}^{\prime}=h_{i}\text{
for }i\in{\mathcal{N}},0<h_{i}^{\prime}\leq 1\text{ for
}i\notin{\mathcal{N}},h^{\prime}\text{ compatible with }A\\}.$ (3.38)

The probability density of
$P^{\Lambda_{N}}_{W,h^{\prime}}(du_{\Lambda_{N}}\,ds_{\Lambda_{N}})$ specified
in (1.16) is dominated by

$\displaystyle
c\,e^{-\frac{1}{2}\sum_{i,j\in\Lambda_{N}}W_{ij}(B_{ij}-1)}e^{-\sum_{i\in\Lambda_{N}}h_{i}(B_{i}-1)}p((e^{u_{i}})_{i\in\Lambda_{N}})\prod_{i\in\Lambda_{N}}\frac{e^{-u_{i}}}{2\pi}$
(3.39)

with some numerical polynomial $p$ with positive coefficients and a constant
$c=c((h_{i})_{i\in{\mathcal{N}}})$, uniformly in $h^{\prime}\in\mathcal{H}$.
This dominating function is integrable by our assumption on the pinning
strength function $h$, cf. remarks below formula (1.6). Hence, by dominated
convergence, we obtain

$\displaystyle\lim_{h^{\prime}\downarrow
h}E^{\Lambda_{N}}_{W,h^{\prime}}[f(u_{\Lambda_{N}},s_{\Lambda_{N}})]=E^{\Lambda_{N}}_{W,h}[f(u_{\Lambda_{N}},s_{\Lambda_{N}})]$
(3.40)

for every bounded measurable function
$f:({\mathbb{R}}^{2})^{\Lambda_{N}}\to{\mathbb{R}}$. The same argument applies
when we replace $\Lambda_{N}$ by $A$ and $h$ by $H|_{A}$. Note that $H|_{A}$
inherits the assumption on the pinning strength function phrased below formula
(1.6). This yields

$\displaystyle\lim_{H^{\prime}\downarrow
H}E^{A}_{W,H^{\prime}}[f(u_{A},s_{A})]=E^{A}_{W,H}[f(u_{A},s_{A})]$ (3.41)

for every bounded measurable function
$f:({\mathbb{R}}^{2})^{A}\to{\mathbb{R}}$. The left-hand sides of (3.40) and
(3.41) agree for $h^{\prime}\in\mathcal{H}$ and the induced $H^{\prime}$,
which allows us to conclude.

The first claim in Corollary 2.3 is an immediate consequence since
$(e^{u_{j}},s_{j}e^{u_{j}})_{j\in B}$ uniquely determines $(u_{j},s_{j})_{j\in
B}$ and $\mathcal{B}_{j}=\\{j\\}$ holds for $j\in B=A\cap\Lambda_{N}$.

##  4 Tightness

To prove Theorem 2.1 we use a slight reformulation of a bound from Lemma 4 in
[DSZ10]. This will be an essential ingredient of the argument. Recall the
abbreviation $B_{ij}$ introduced in (1.14).

######  Lemma 4.1 (Alignment of two spins)

Consider the $H^{2|2}$-model on a finite vertex set $\Lambda$ with arbitrary
weights $W$ and pinning $h$. For $W_{ij}>0$ and $\delta>W_{ij}^{-1}$, the
following bound holds

$\displaystyle P^{\Lambda}_{W,h}(B_{ij}\geq 1+\delta)\leq W_{ij}\delta
e^{-(W_{ij}\delta-1)}.$ (4.1)

Proof. The proof is based on Lemmas 3 and 4 in [DSZ10], which generalize to
non-constant weights $W_{ij}$ in a straightforward way. Using first
Chebyshev’s inequality and then Lemma 3 in [DSZ10], we obtain for all
$\delta>0$ and all $\gamma\in(0,1)$

$\displaystyle P^{\Lambda}_{W,h}(B_{ij}\geq
1+\delta)=E^{\Lambda}_{W,h}[1_{\\{B_{ij}\geq 1+\delta\\}}]\leq
E^{\Lambda}_{W,h}[e^{W_{ij}\gamma(B_{ij}-1-\delta)}]\leq(1-\gamma)^{-1}e^{-W_{ij}\gamma\delta}.$
(4.2)

The upper bound is of the form $e^{-f(\gamma)}$ with
$f(\gamma)=\log(1-\gamma)+\alpha\gamma$ and $\alpha=W_{ij}\delta$. In order to
optimize this upper bound, we maximize $f$. Observe that
$f^{\prime}(\gamma)=\alpha-(1-\gamma)^{-1}$, which equals zero iff
$\gamma=\gamma_{c}=1-\alpha^{-1}$. By assumption $\alpha>1$, and hence the
critical point $\gamma_{c}$ belongs to $(0,1)$. Using
$f(1-\alpha^{-1})=-\log\alpha+\alpha-1$, the claim follows.

Using Lemma 4.1 and the reduction to the effective model, we prove our main
result.

Proof of Theorem 2.1. We treat the two cases, uniform pinning (“U”) and
pinning at one point (“1P”) simultaneously.

Let $\rho>0$. It suffices to find $M=M(\rho)>0$ such that for all
$N\in{\mathbb{N}}$, $\epsilon>0$, and $p,q\in\Lambda_{N}$ with $p\neq q$, one
has

case U: $\displaystyle P^{\Lambda_{N}}_{W,\epsilon 1}(|u_{p}-u_{q}|\geq
M)\leq\rho,$ (4.3) case 1P: $\displaystyle
P^{\Lambda_{N}}_{W,\epsilon\delta_{p}}(|u_{p}-u_{q}|\geq M)\leq\rho.$ (4.4)

Note that in (4.4) the vertex $p$ appears twice, as pinning point and in the
variable $u_{p}$. This suffices indeed because for $i_{0},p,q\in\Lambda_{N}$
we have the estimate

$\displaystyle P^{\Lambda_{N}}_{W,\epsilon\delta_{i_{0}}}(|u_{p}-u_{q}|\geq
M)\leq
P^{\Lambda_{N}}_{W,\epsilon\delta_{i_{0}}}\left(|u_{i_{0}}-u_{p}|\geq\frac{M}{2}\right)+P^{\Lambda_{N}}_{W,\epsilon\delta_{i_{0}}}\left(|u_{i_{0}}-u_{q}|\geq\frac{M}{2}\right).$
(4.5)

Let $N\in{\mathbb{N}}$, $\epsilon>0$, and $p,q\in\Lambda_{N}$ with $p\neq q$.
Recall the notation introduced at the beginning of Section 2.1. We abbreviate
$d=d(p,q)\geq 1$. Let $\overline{0}:=1$, $\overline{1}:=0$. For
$j=(j_{0},\ldots,j_{|j|-1})\in{\mathcal{T}}^{N}\setminus\\{\emptyset\\}$, let
$\overline{j}:=(\overline{j}_{0},j_{1},j_{2},\ldots,j_{|j|-1})$ be the nearest
neighbor of $j$ on the same level. We consider the set
$A\subseteq{\mathcal{T}}^{N}$ given by

$\displaystyle A:=$
$\displaystyle\\{p,q\\}\cup\\{j\in{\mathcal{T}}^{N}\setminus\\{\emptyset\\}:j\not\preceq
p,j\not\preceq q,({\tt cut}(j)\preceq p\text{ or }{\tt cut}(j)\preceq q)\\}$
(4.6) $\displaystyle=$ $\displaystyle\\{p,q\\}\cup\\{\overline{{\tt
cut}^{l}(p)}:0\leq l\leq N-1,l\neq d-1\\}\cup\\{\overline{{\tt
cut}^{l}(q)}:0\leq l\leq d-2\\}.$

An illustration is given in Figure 3.

Figure 3: The vertices $p,q\in\Lambda_{4}$ have hierarchical distance $d=3$.
The maximal antichain from (4.6) equals
$A:=\\{p,q,\overline{p},\overline{q},\overline{{\tt cut}(p)},\overline{{\tt
cut}(q)},\overline{{\tt cut}^{3}(p)}\\}$. The path $\pi$ in (4.11) is given by
$\pi=(p,\overline{{\tt cut}(p)},\overline{{\tt cut}(q)},q)$.

Note that $A$ is a maximal antichain. Furthermore, it is compatible with both
pinning functions, $h=\epsilon 1$ (case U) and $h=\epsilon\delta_{p}$ (case
1P). We remark that this is the reason why we need $p$ as a pinning point in
formula (4.4).

We consider the path $\pi=(\pi_{l})_{0\leq l\leq m}$ in the antichain $A$ from
$\pi_{0}=p$ to $\pi_{m}=q$ given by the concatenation of $p$, $(\overline{{\tt
cut}^{l}(p)})_{1\leq l\leq d-2}$, the reversed path of $(\overline{{\tt
cut}^{l}(q)})_{1\leq l\leq d-2}$, and $q$. In other words, for $m=d=1$, we
have $\pi=(p,q)$ and for $d\geq 2$, we have $m=2d-3$, and

$\displaystyle\pi_{l}=\left\\{\begin{array}[]{ll}p&\text{for }l=0,\\\
\overline{{\tt cut}^{l}(p)}&\text{for }1\leq l\leq d-2,\\\ \overline{{\tt
cut}^{m-l}(q)}&\text{for }d-1\leq l\leq m-1,\\\ q&\text{for
}l=m.\end{array}\right.$ (4.11)

Note that we exclude on purpose the points $\overline{p}=\overline{{\tt
cut}^{0}(p)}$ and $\overline{q}=\overline{{\tt cut}^{0}(q)}$ in the second and
third line of (4.11) (though they belong to the antichain) since they do not
help in the estimates below.

In the case $d\geq 2$, the edges $\\{\pi_{l},\pi_{l+1}\\}$ (with
$l=0,\ldots,m-1$) of the path $\pi$ have the weights

$\displaystyle\beta_{l}^{\prime}:=W_{\pi_{l}\pi_{l+1}}=$ $\displaystyle
2^{\ell(\pi_{l})+\ell(\pi_{l+1})}w(\ell(\pi_{l}\wedge\pi_{l+1}))$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}2^{2l+1}w(l+2)=\beta_{l}&\text{if
}0\leq l\leq d-3,\\\ 2^{2d-4}w(d)=\frac{1}{2}\beta_{d-2}&\text{if }l=d-2,\\\
2^{2(m-l)-1}w(m-l+1)=\beta_{m-l-1}&\text{if }d-1\leq l\leq
m-1.\end{array}\right.$ (4.15)

In the case $d=1$, the single edge $\\{p,q\\}$ of the path $\pi$ has the
weight $\beta_{0}^{\prime\prime}:=W_{pq}=w(1)$. We choose
$\delta_{0}^{\prime\prime}>0$ large enough such that
$\beta_{0}^{\prime\prime}\delta_{0}^{\prime\prime}>1$ and
$\beta_{0}^{\prime\prime}\delta_{0}^{\prime\prime}e^{-(\beta_{0}^{\prime\prime}\delta_{0}^{\prime\prime}-1)}<\rho$.
We show below the following claim.
Claim: There is a sequence $(\delta_{l})_{l\in{\mathbb{N}}_{0}}$ of positive
numbers (independent of $p$ and $q$) such that

$\displaystyle\beta_{l}\delta_{l}>1\text{ for all
}l,\quad\sum_{l=0}^{\infty}\operatorname{arcosh}(1+\delta_{l})<\infty\quad\mbox{and}\quad\sum_{l=0}^{\infty}\beta_{l}\delta_{l}e^{-\beta_{l}\delta_{l}+1}<\frac{\rho}{2}.$
(4.16)

Using this claim, we show first that the estimates (4.3) and (4.4) hold. We
define

$\displaystyle M:=\max\Big{\\{}\sup_{d^{\prime}\geq
2}\Big{(}2\sum_{l=0}^{d^{\prime}-3}\operatorname{arcosh}(1+\delta_{l})+\operatorname{arcosh}(1+2\delta_{d^{\prime}-2})\Big{)},\operatorname{arcosh}(1+\delta_{0}^{\prime\prime})\Big{\\}}.$
(4.17)

As a consequence of (4.16), the sequence $(\delta_{l})_{l}$ is bounded and $M$
is finite.

We consider now the case $d\geq 2$. We introduce

$\displaystyle\delta_{l}^{\prime}=$
$\displaystyle\left\\{\begin{array}[]{ll}\delta_{l}&\text{for }0\leq l\leq
d-3,\\\ 2\delta_{d-2}&\text{for }l=d-2,\\\ \delta_{m-l-1}&\text{for }d-1\leq
l\leq m-1.\end{array}\right.$ (4.21)

Note that

$\displaystyle\sum_{l=0}^{m-1}\operatorname{arcosh}(1+\delta_{l}^{\prime})=2\sum_{l=0}^{d-3}\operatorname{arcosh}(1+\delta_{l})+\operatorname{arcosh}(1+2\delta_{d-2})\leq
M.$ (4.22)

Since $A$ is a maximal antichain containing $p$ and $q$, the distributions of
$u_{p}-u_{q}$ with respect to $P^{\Lambda_{N}}_{W,h}$ and $P^{A}_{W,H}$
coincide by Corollary 2.3; here $H$ is the extension of the pinning function
$h$ given in (2.10). Using (4.22) in the first inequality, we estimate

$\displaystyle P^{\Lambda_{N}}_{W,h}(|u_{p}-u_{q}|\geq M)=$ $\displaystyle
P^{A}_{W,H}(|u_{\pi_{0}}-u_{\pi_{m}}|\geq M)$ $\displaystyle\leq$
$\displaystyle
P^{A}_{W,H}\Big{(}|u_{\pi_{0}}-u_{\pi_{m}}|\geq\sum_{l=0}^{m-1}\operatorname{arcosh}(1+\delta_{l}^{\prime})\Big{)}$
$\displaystyle\leq$ $\displaystyle
P^{A}_{W,H}\Big{(}\sum_{l=0}^{m-1}|u_{\pi_{l}}-u_{\pi_{l+1}}|\geq\sum_{l=0}^{m-1}\operatorname{arcosh}(1+\delta_{l}^{\prime})\Big{)}$
$\displaystyle\leq$
$\displaystyle\sum_{l=0}^{m-1}P^{A}_{W,H}(|u_{\pi_{l}}-u_{\pi_{l+1}}|\geq\operatorname{arcosh}(1+\delta_{l}^{\prime}))$
$\displaystyle=$
$\displaystyle\sum_{l=0}^{m-1}P^{A}_{W,H}(\cosh(u_{\pi_{l}}-u_{\pi_{l+1}})\geq
1+\delta_{l}^{\prime}).$ (4.23)

Recall the notation $B_{ij}$ from formula (1.14). In particular,
$\cosh(u_{i}-u_{j})\leq B_{ij}$ for $i,j\in A$. To estimate the $l$-th summand
on the right-hand side of formula (4.23) we observe that
$W_{\pi_{l}\pi_{l+1}}=\beta_{l}^{\prime}$ and
$\beta_{l}^{\prime}\delta_{l}^{\prime}=\beta_{l}\delta_{l}>1$ as a consequence
of (4.15), (4.21), and Claim (4.16) so that Lemma 4.1 is applicable and yields

$\displaystyle P^{A}_{W,H}(\cosh(u_{\pi_{l}}-u_{\pi_{l+1}})\geq
1+\delta_{l}^{\prime})\leq P^{A}_{W,H}(B_{\pi_{l}\pi_{l+1}}\geq
1+\delta_{l}^{\prime})\leq\beta_{l}^{\prime}\delta_{l}^{\prime}e^{-(\beta_{l}^{\prime}\delta_{l}^{\prime}-1)}.$
(4.24)

Summing over $l$ and using Claim (4.16) in the last step, we conclude

$\displaystyle P^{\Lambda_{N}}_{W,h}(|u_{p}-u_{q}|\geq
M)\leq\sum_{l=0}^{m-1}\beta_{l}^{\prime}\delta_{l}^{\prime}e^{-(\beta_{l}^{\prime}\delta_{l}^{\prime}-1)}$
$\displaystyle=$ $\displaystyle
2\sum_{l=0}^{d-3}\beta_{l}\delta_{l}e^{-(\beta_{l}\delta_{l}-1)}+\beta_{d-2}\delta_{d-2}e^{-(\beta_{d-2}\delta_{d-2}-1)}\leq
2\sum_{l=0}^{\infty}\beta_{l}\delta_{l}e^{-(\beta_{l}\delta_{l}-1)}<\rho.$
(4.25)

For $d=1$, using $\operatorname{arcosh}(1+\delta_{0}^{\prime\prime})\leq M$
from (4.17) and again Lemma 4.1, it also holds

$\displaystyle P^{\Lambda_{N}}_{W,h}(|u_{p}-u_{q}|\geq M)\leq
P^{\Lambda_{N}}_{W,h}(\cosh(u_{p}-u_{q})\geq
1+\delta_{0}^{\prime\prime})\leq\beta_{0}^{\prime\prime}\delta_{0}^{\prime\prime}e^{-(\beta_{0}^{\prime\prime}\delta_{0}^{\prime\prime}-1)}<\rho.$
(4.26)

It remains to show Claim (4.16). Assume that the hypothesis (2.5) of Theorem
2.1 holds. To ensure
$\sum_{l=0}^{\infty}\operatorname{arcosh}(1+\delta_{l})<\infty$, it is
necessary that $\delta_{l}\to 0$ as $l\to\infty$. On the other hand, to ensure
$\sum_{l=0}^{\infty}\beta_{l}\delta_{l}e^{-\beta_{l}\delta_{l}+1}<\frac{\rho}{2}$,
we will choose $\delta_{l}$ such that $\beta_{l}\delta_{l}\to\infty$ fast
enough as $l\to\infty$ and $\beta_{l}\delta_{l}$ is large enough for a
sufficiently large initial piece $l=0,\ldots,l_{1}$. More precisely, we
proceed as follows.

From assumption (2.5) we know $\lim_{l\to\infty}\beta_{l}=\infty$. Take
$l_{0}\in{\mathbb{N}}$ so large that $\log\log\sqrt{\beta_{l}}$ is well-
defined and positive for all $l\geq l_{0}$. For these $l$, we set

$\displaystyle\tilde{\delta}_{l}:=\frac{1}{\beta_{l}}\Big{(}1+\log\sqrt{\beta_{l}}+\frac{3}{2}\log\log\sqrt{\beta_{l}}\Big{)}>\frac{1}{\beta_{l}}.$
(4.27)

We observe that

$\displaystyle\sum_{l=l_{0}}^{\infty}\beta_{l}\tilde{\delta}_{l}e^{-\beta_{l}\tilde{\delta}_{l}+1}=\sum_{l=l_{0}}^{\infty}\Big{(}1+\log\sqrt{\beta_{l}}+\frac{3}{2}\log\log\sqrt{\beta_{l}}\Big{)}\exp\Big{(}-\log\sqrt{\beta_{l}}-\frac{3}{2}\log\log\sqrt{\beta_{l}}\Big{)}$
$\displaystyle=$
$\displaystyle\sum_{l=l_{0}}^{\infty}\Big{(}1+\log\sqrt{\beta_{l}}+\frac{3}{2}\log\log\sqrt{\beta_{l}}\Big{)}\frac{1}{\sqrt{\beta_{l}}}(\log\sqrt{\beta_{l}})^{-\frac{3}{2}}<\infty.$
(4.28)

The last estimate used assumption (2.5). However, we remark that its full
strength is not used here, but only below. The estimate (4.28) implies that
there exists $l_{1}\geq l_{0}$ such that

$\displaystyle\sum_{l=l_{1}}^{\infty}\beta_{l}\tilde{\delta}_{l}e^{-\beta_{l}\tilde{\delta}_{l}+1}<\frac{\rho}{4}.$
(4.29)

We choose $\delta_{l}>\beta_{l}^{-1}$ for $0\leq l\leq l_{1}-1$ large enough,
such that

$\displaystyle\sum_{l=0}^{l_{1}-1}\beta_{l}\delta_{l}e^{-\beta_{l}\delta_{l}+1}<\frac{\rho}{4}.$
(4.30)

For $l\geq l_{1}$, we set $\delta_{l}:=\tilde{\delta}_{l}$. Then, the
condition for the second series in (4.16) follows. Observe that
$\beta_{l}\delta_{l}>1$ for all $l\in{\mathbb{N}}_{0}$ by (4.27) and our
choice of $\delta_{l}$ for small $l$.

It remains to prove the inequality for the first series in (4.16), which is
equivalent to the claim

$\displaystyle\sum_{l=l_{1}}^{\infty}\operatorname{arcosh}(1+\delta_{l})<\infty.$
(4.31)

We estimate the $l$-th summand in the last series:

$\displaystyle\operatorname{arcosh}(1+\delta_{l})\leq$
$\displaystyle\sqrt{2\delta_{l}}=\sqrt{\frac{2}{\beta_{l}}}\sqrt{1+\log\sqrt{\beta_{l}}+\frac{3}{2}\log\log\sqrt{\beta_{l}}}=O\left(\sqrt{\frac{\log\beta_{l}}{\beta_{l}}}\right)$
(4.32)

as $l\to\infty$, which is summable over $l$ by hypothesis (2.5). This
completes the proof of Claim (4.16) and hence the proof of the theorem.

###### Remark 4.2

In the proof of Theorem 2.1 it is shown that for $p,q\in\Lambda_{N}$ with
$d(p,q)=d$, the set

$\displaystyle A:=\\{p,q\\}\cup\\{\overline{{\tt cut}^{l}(p)}:0\leq l\leq
N-1,l\neq d-1\\}\cup\\{\overline{{\tt cut}^{l}(q)}:0\leq l\leq d-2\\}$ (4.33)

is a maximal antichain containing $p$ and $q$; see Figure 3 for an
illustration. In order to study the distribution of $(u_{p},u_{q})$ with
respect to $P^{\Lambda_{N}}_{W,h}$ with a pinning function $h$ such that $A$
is compatible with it, cf. Figure 2, one can equally well study its
distribution with respect to the reduced model $P^{A}_{W,H}$. Note that
$|\Lambda_{N}|=2^{N}$, whereas $|A|=N+d$, resulting in an exponential decrease
of complexity. More generally, one can construct a maximal antichain $A(B)$
containing a finite set $B\subseteq\Lambda_{N}$ as follows. The set $A(B)$
consists of $B$ and all vertices which are minimal elements with respect to
$\preceq$ in the set $\\{i\in{\mathcal{T}}^{N}:i\not\preceq j\text{ for all
}j\in B\\}$. For fixed $|B|$, as $N\to\infty$, the size of $A(B)$ is again
bounded by $O(N)$.

##  Appendix A Appendix: Grassmann algebras and supersymmetry operators

#### Grassmann algebras.

Here we give some more details on the Grassmann algebra $\mathcal{A}$
underlying the $H^{2|2}$-model with vertex set $\Lambda$. We consider a real
vector space $V$ of finite dimension $K$. We assume that $V$ has a basis
$\mathcal{B}=\\{\chi_{1},\ldots,\chi_{K}\\}$ consisting of $\xi_{i}$,
$\eta_{i}$ for $i\in\Lambda$ and possibly additional Grassmann parameters.
Then the Grassmann algebra $\mathcal{A}$ is defined as the direct sum of
exterior product spaces $\mathsf{\Lambda}^{k}V$ of $V$:
$\mathcal{A}=\bigoplus_{k=0}^{K}\mathsf{\Lambda}^{k}V=\mathcal{A}_{0}\oplus\mathcal{A}_{1}$
with the even (commuting) subalgebra $\mathcal{A}_{0}=\bigoplus_{k=0}^{\lfloor
K/2\rfloor}\mathsf{\Lambda}^{2k}V$ and the odd (anticommuting) subspace
$\mathcal{A}_{1}=\bigoplus_{k=0}^{\lfloor(K-1)/2\rfloor}\mathsf{\Lambda}^{2k+1}V$.
Note that the body of an even element is the projection to the $0$th-
component:
$\operatorname{body}:\mathcal{A}_{0}\to\mathsf{\Lambda}^{0}V={\mathbb{R}}$.
Every element $\omega$ in the Grassmann algebra $\mathcal{A}$ can be uniquely
written as follows:

$\displaystyle\omega=\sum_{I\subseteq\\{1,\ldots,K\\}}c_{I}\chi^{I},$ (A.1)

with coefficients $c_{I}\in{\mathbb{R}}$ and basis elements
$\chi^{I}:=\chi_{i_{1}}\ldots\chi_{i_{l}}$ for $I=\\{i_{1},\ldots,i_{l}\\}$
with $i_{1}<\cdots<i_{l}$ and $\chi^{\emptyset}=1$.

#### Grassmann derivatives.

For every basis element $\chi\in\mathcal{B}$ and every element
$\omega\in\mathcal{A}$, we have a unique decomposition
$\omega=\omega_{0}+\chi\omega_{1}$, where $\omega_{0}$ and $\omega_{1}$ belong
to the subalgebra of $\mathcal{A}$ generated by the elements of
$\mathcal{B}\setminus\\{\chi\\}$. We define the Grassmann derivative

$\displaystyle\partial_{\chi}\omega:=\omega_{1}.$ (A.2)

Note that this derivative depends not only on $\chi$ and $\omega$, but also on
the choice of the basis $\mathcal{B}$. This is in analogy to the ordinary
partial derivative, which also depends on the choice of other variables which
are kept fixed.

#### Supersymmetry operators.

We review now the (super)symmetry operators of the $H^{2|2}$-model. For
$n\in{\mathbb{N}}$, let $[n]:=\\{1,2,\ldots,n\\}$. We define the parity
function

$\displaystyle p:[5]\to\\{0,1\\},\quad
p(i)=\left\\{\begin{array}[]{ll}0&i\in[3],\\\
1&i\in\\{4,5\\}.\end{array}\right.$ (A.5)

For $i,j\in[5]$, we set

$\displaystyle\sigma_{ij}:=(-1)^{p(i)+p(j)+p(i)p(j)}=\left\\{\begin{array}[]{ll}1&i,j\in[3],\\\
-1&\text{otherwise.}\end{array}\right.$ (A.8)

Using the matrix

$\displaystyle
g=(g_{ij})_{i,j\in[5]}:=\left(\begin{array}[]{rrr|rr}1&0&0&0&0\\\ 0&1&0&0&0\\\
0&0&-1&0&0\\\ \hline\cr 0&0&0&0&1\\\ 0&0&0&-1&0\end{array}\right)$ (A.14)

the inner product of $v=(x,y,z,\xi,\eta)$ and
$v^{\prime}=(x^{\prime},y^{\prime},z^{\prime},\xi^{\prime},\eta^{\prime})$
reads

$\displaystyle\left\langle{v^{\prime},v}\right\rangle=v^{\prime}gv^{t}.$
(A.15)

We write $v=(v^{i})_{i\in[5]}$ and $\partial_{v^{i}}=\frac{\partial}{\partial
v^{i}}$.

######  Definition A.1 (Supersymmetry operators)

For $i,j\in[5]$, we define the supersymmetry operators

$\displaystyle
L_{ij}=L_{ij}^{v}:=\sum_{k\in[5]}[v^{k}g_{kj}\partial_{v^{i}}-v^{k}\sigma_{ij}g_{ki}\partial_{v^{j}}]$
(A.16)

We observe that $L_{ij}=-\sigma_{ji}L_{ji}$ for all $i,j$. Furthermore,
$L_{11}=L_{22}=L_{33}=0$,

  * •

$L_{12}=y\partial_{x}-x\partial_{y}$ generates Euclidean rotations in the
$x$-$y$-plane,

  * •

$-L_{13}=z\partial_{x}+x\partial_{z}$ and
$-L_{23}=z\partial_{y}+y\partial_{z}$ generate Lorentz boosts in the $x$-$z$\-
and $y$-$z$-plane, respectively,

  * •

$L_{44}=-2\eta\partial_{\xi}$, $L_{45}=\xi\partial_{\xi}-\eta\partial_{\eta}$,
and $L_{55}=2\xi\partial_{\eta}$ are even operators, and

  * •

$L_{14}=x\partial_{\xi}-\eta\partial_{x}$,
$L_{24}=y\partial_{\xi}-\eta\partial_{y}$,
$L_{15}=\xi\partial_{x}+x\partial_{\eta}$,
$L_{25}=\xi\partial_{y}+y\partial_{\eta}$,
$L_{34}=-(\eta\partial_{z}+z\partial_{\xi})$, and
$L_{35}=\xi\partial_{z}-z\partial_{\eta}$ are odd operators.

We define

$\displaystyle
Q^{v}:=L_{15}-L_{24}=x\partial_{\eta}-y\partial_{\xi}+\xi\partial_{x}+\eta\partial_{y}.$
(A.17)

We collect some basic facts on these supersymmetry operators. For more details
see Section 4 and Appendices B and C of [DSZ10].

######  Fact A.2 (Consequences of supersymmetry)

  1. 1.

For $i,j\in[5]$, the following holds.

    1. (a)

The operator $L_{ij}$ annihilates the inner product as follows

$\displaystyle(L_{ij}^{v}+L_{ij}^{v^{\prime}})\left\langle{v,v^{\prime}}\right\rangle=0\quad\text{for
all }v,v^{\prime}\in H^{3|2}.$ (A.18)

Moreover, for $k\in{\mathbb{N}}$ and any smooth superfunction
$f:\mathcal{A}_{0}^{k^{2}}\to\mathcal{A}_{0}$, as $v_{1},\ldots,v_{k}$ run
over $H^{3|2}$, one has

$\displaystyle\sum_{l=1}^{k}L_{ij}^{v_{l}}f((\left\langle{v_{j},v_{j^{\prime}}}\right\rangle)_{1\leq
j,j^{\prime}\leq k})=0.$ (A.19)

    2. (b)

_Ward identity._ For any smooth superfunction
$f:(H^{3|2}_{+})^{k}\to\mathcal{A}_{0}$ with $k\in{\mathbb{N}}$ such that $f$
is fast decaying on $(H^{2|2})^{k}$, one has

$\displaystyle\int_{(H^{2|2})^{k}}\mathcal{D}v_{1}\ldots\mathcal{D}v_{k}\,\sum_{l=1}^{k}L_{ij}^{v_{l}}f(v_{1},\ldots,v_{k})=0.$
(A.20)

  2. 2.

The operator $Q$ annihilates the following inner products

$\displaystyle(Q^{v}+Q^{v^{\prime}})\left\langle{v,v^{\prime}}\right\rangle=0,\quad
Q^{v}\left\langle{v,o}\right\rangle=0,\quad\text{for all }v,v^{\prime}\in
H^{3|2}.$ (A.21)

#### Acknowledgements.

The authors were supported by the DFG Priority Programme 2265 Random Geometric
Systems.

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