Random clusters in the Villain and XY models Julien Dub edatHugo Falconet November 30 2022

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Random clusters in the Villain and XY models

Julien Dub´edat ∗Hugo Falconet †

November 30, 2022

Abstract

In the Ising and Potts model, random cluster representations provide a geometric

interpretation to spin correlations. We discuss similar constructions for the Villain and

XY models, where spins take values in the circle, as well as extensions to models with

diﬀerent single site spin spaces. In the Villain case, we highlight natural interpretation

in terms of the cable system extension of the model. We also list questions and open

problems on the cluster representations obtained in this fashion.

1 Introduction

The planar rotator model, or XY model, is a continuous spin model where single site spins

take values in the unit circle Uwhich exhibits the Berezinskii-Kosterlitz-Thouless phase

transition [8, 33], a fact proved ﬁrst by Fr¨ohlich and Spencer [25] and revisited more recently

in [32, 40, 1]. This result states that above a critical temperature Tc, the decay of spin

correlation is exponential, whereas there is only a power law decay below Tc, i.e. hσxσyiT=

|x−y|−αT+o(1) for some exponent αT>0 whenever T < Tc. Moreover, in this low temperature

phase it is conjectured since the work of Fr¨ohlich and Spencer [26] that the the XY model

should behave at large scale like eiTeﬀ Φ, where Φ is the planar Gaussian free ﬁeld (GFF).

The Villain model is a variant with the same phenomenology which is more tractable

due to exact duality identities involving the integer-valued GFF. In this article, the Villain

interaction is singled for a related but distinct reason, namely its relation with the so-called

cable systems: there is a natural way to extend the spins deﬁned on the vertices of the lattice

to a continuous family of spins on the edges with a locally Brownian structure. Similar

constructions were used in the context of isomorphism theorems between the Gaussian free

ﬁeld and the occupation ﬁeld of trajectories by Lupu in [34], and then in the context of the

ﬁrst passage sets of the 2dGFF in [2], the set of points in the domain that can be connected

to the boundary by a path along which the GFF is greater than or equal to a ﬁxed height.

∗Department of Mathematics, Columbia University, 2990 Broadway, New York, NY 10027, USA.

†Courant Institute, New York University, 251 Mercer street, New York, NY 10012, USA.

arXiv:2210.03620v2 [math.PR] 28 Nov 2022

Our goal is to provide a geometric interpretation to spin correlations: ﬁrst, for the Villain

model in Proposition 3.2 and Proposition 3.3 below, by introducing random clusters in the

extended Villain model, and then, for appropriate random cluster constructions in the XY

model, which is the content of Theorem 4.5 below. In fact, this theorem covers more generally

the O(2) model, the XY and Villain models being special cases. Our constructions have the

same spirit as the one of the Fortuin-Kasteleyn (FK)/random cluster representations of the

Ising and Potts models (the joint distribution of random clusters with spins goes under

the name of Edwards-Sokal coupling [21] [30, Section 1.4]). In the Ising model, two equal

nearest neighbor spins are in the same cluster with a ﬁxed probability, independently of other

edges and the spins themselves can be retrieved from the geometrical clusters by attributing

random signs to each cluster [30, Section 1.4]. Then, [30, Theorem 1.6] states

hσxσyiTcP(x↔y),

where in the right-hand side xis connected to yif they are in the same cluster. In fact,

they are equal up to a multiplicative constant, this doesn’t speciﬁcally rely on being at the

critical temperature and the same holds for the Potts model. Our motivation is twofold: ﬁrst,

akin to the FK model, these random clusters can be seen as a tool to study the continuous

spin models; second, the FK model is a model interesting in itself (it provides a continuous

interpolation of the q-Potts model in which qis restricted to be an integer) and we believe

the same is true in our case.

The planar Ising model, with spins taking values only in {+1,−1}has a phenomenology

diﬀerent than the one of the aforementioned continuous spin models. Rather, it is only at

the critical temperature that the spin correlations have polynomial decay and below that

temperature, the two-point correlation function may not vanish (as the separation goes to

inﬁnity), depending on the boundary conditions/Gibbs measure considered. Indeed, in this

case and at low temperature, there is no uniqueness of translation invariant inﬁnite-volume

measure. Although we expect qualitative similarities, the interfaces of our random clusters

in the KT phase and the interfaces in the critical Ising/FK model may also be diﬀerent, and

this is discussed in further details below.

The Swendsen-Wang (SW) algorithm is commonly used to implement Monte Carlo simu-

lations of the Ising spin model; at each step an FK cluster of spins (sampled conditionally on

the spins) is ﬂipped. More precisely, a step of the algorithm consists of the following: ﬁrst,

a bond is formed between every pair of nearest neighbors that are equal, with an explicit

probability pdepending on the temperature and coupling constant of the model. Then, all

spins in each cluster are ﬂipped. Finally, all bonds are erased and the step is complete. The

algorithm generalizes to the q-state Potts models in which each lattice site corresponds to

a variable that can take qdiﬀerent values. One of these qvalues is assigned with probabil-

ity 1/q to each cluster and all variables in the cluster take this new value. In [43], when

considering continuous spin models, Wolﬀ replaced the global spin-inversion operation by a

reﬂection operation in which only the connected component of one spin chosen uniformly at

random is reﬂected over a randomly oriented plane. At each iteration, a new spin and an

orientation of the plane are chosen. A review of cluster algorithms in a general framework

(which includes Potts model, random surfaces, continuous spins, among others) can be found

in Section 2 of [17].

In the case of the XY model, Chayes considered a graphical representation based on the

Wolﬀ algorithm in [14]. He did so by writing spins in the form (σxcos θx, τxsin θx) where

σx, τx∈ {−1,1}and then by considering the FK-Ising model associated with one of these ±1

spins. In particular, he proved that these percolation measures satisfy the FKG property and

obtained a characterization of the positive magnetization of the system in term of percolation

in the graphical representation (see also [11] for an extension to the Heisenberg model). The

clusters he considered are the same as those in Lemma 4.1. In this work, in order to obtain

result for higher spin observables, we also consider clusters made of correlated bonds (instead

of only one bond as in [14, 11, 17]).

To conclude this introduction, we mention that the planar Villain and XY models have

recently been the center of a regain of interest in the mathematics community, in particular

with the the works [35, 42, 27, 28] and the ones mentioned above. Furthermore, some progress

was made on the Fr¨ohlich Spencer conjecture as [5] proved the convergence of the integer-

valued Gaussian free ﬁeld towards the Gaussian free ﬁeld, when the temperature is suﬃciently

large (which corresponds to low temperature continuous spin model). Ultimately, we refer

the reader who wish to learn more about these spin models to [24, 36], for an introduction

to the ﬁeld.

This paper is organized as follows. First, we deﬁne the Villain model and cable systems in

Section 2. Then, in Section 3 we prove some two sided estimates for some spin observables in

terms of connectivity properties of random clusters. In Section 4, we generalize these results

to O(2) models by introducing random clusters made of open/close bonds with appropriate

joint distributions. In Section 5.1, we explain in which sense the usual random cluster model

and its dilute version (related with the Blume-Capel-Potts model) can be seen from random

clusters associated with similar cable systems but with a diﬀerent continuous-time Markov

chain. Finally, we list questions and open problems in Section 5.2.

Acknowledgments. We wish to thank Ron Peled for pointing out several references.

2 Villain model and cable systems

Villain model. The Villain model is a statistical mechanics model with spins taking values

in the unit circle U'R/2πZand interactions given by a θfunction. The speciﬁc choice of

interaction was motivated by the following considerations. If one considers a low temperature

T(or βlarge), then the XY weight eβcos(θx−θy)can be approximated by e−1

2β(θx−θy)2as the

spins tend to be aligned in this regime, by a Taylor expansion. In order to preserve a

model well-deﬁned for angles modulo 2π, it is natural to consider a periodized version of this

interaction, leading to the Villain model. The reason one is interested in this approximation

comes from the fact that the Fourier modes of this new interaction are Gaussian terms, which

can be used to show powerful identities relating Villain observables to dual, integer-valued

height models.

More precisely, let G= (V, E) be a ﬁnite subgraph of Zd(typically, a box or a torus). The

probability measure on conﬁgurations (ux)x∈V= (eiθx)x∈V∈UVis given by 1

Ze−H(θ)Qx∈Vdθx

where the energy is deﬁned as

e−H(θ)=Y

(xy)∈EX

n∈Z

exp −β(θx−θy+ 2nπ)2.

Here, β > 0 plays the role of inverse temperature.

We now turn to the Fourier decomposition of this interaction. To this end, recall the

Jacobi Θ function (=(τ)>0)

Θ(z, τ) = X

n∈Z

eiπn2τe2iπnz,

so that (for t= 4πβ > 0, v= (θx−θy)/2π)

Θ(itv, it) = eπtx2X

n∈Z

exp(−πt(v+n)2).

We have the functional equation (e.g. (8.9) in [13]) √tΘ(z, it) = e−πz2

tΘ( z

it , it−1) or

√tX

n∈Z

exp(−πt(x+n)2) = X

n∈Z

e−πn2t−1exp(2iπnx)

= 1 + 2 ∞

n=1

e−πn2t−1cos(2πnx),(2.1)

a standard instance of the Poisson summation formula. Note that the LHS (resp RHS) series

converges faster when t > 1 (resp. t < 1).

Cable systems. While the Villain model was introduced for its special duality properties,

we are exploiting here an indirectly related property. Observe that

pt(θ1, θ2) = 1

√2πt X

n∈Z

exp −1

2t(θ1−θ2+ 2nπ)2(2.2)

where (pt) denotes the transition function for Brownian motion on U'R/2πZ. In particular,

by Chapman-Kolmogorov:

pt+s(θ1, θ3) = Zpt(θ1, θ2)ps(θ2, θ3)dθ2.

Then, the energy can be written as a product of transition functions, i.e.

e−H(θ)= (2πt)|V|/2Y

(xy)∈E

pt(θx, θy),(2.3)

with 1/2t=β(tis thus a temperature parameter).

Now, we consider the cable system (also known as a metric graph) ˜

Gobtained by adding

a segment of length t(e) = tbetween xand y, where e= (xy) is any edge of G(i.e., any

abstract edge in Gbecomes a line segment in ˜

G). One can extend the Villain model from

a measure on UVto a measure on C0(˜

G,U) by sampling a U-valued Brownian bridge (of

duration t, from θxto θy) to ﬁll the values on these added intervals. This is modeled on

the construction of the Gaussian Free Field (GFF) on cable systems by Lupu [34] (where

the single site spin space is R, rather than Uhere). Earlier references on diﬀusions in cable

systems include [6, 22, 23].

More generally, the construction works if the spin space is, say, a compact manifold S

with a Riemannian metric whose associated volume form is µand (pt) is the heat kernel for

a symmetric diﬀusion on that manifold w.r.t. µ. One can also consider Snon-compact if µ

is ﬁnite; if the underlying graph Gis ﬁnite, there is no restriction. The interaction is still

given by Q(xy)∈Ept(ux, uy), the symmetry assumption implies pt(ux, uy) = pt(uy, ux) and (pt)

satisﬁes

pt+s(ux, uy) = ZS

pt(ux, uz)ps(uz, uy)µ(duz).(2.4)

This property is used extend the distribution on vertices to the reﬁned graphs with Kol-

mogorov extension (e.g, the midpoints of edges can be added by using pt/2for the interaction

on each half-edge), and the consistency follows from (2.4). In the case of the GFF, ptis the

heat kernel on Rconsidered by Lupu and in the case of the Villain model described above,

it is the heat kernel on U.

The resulting extended model (Xv)v∈˜

Gprovides a continuous stochastic process with

sample paths in C0(˜

G, S) which satisﬁes a simple and strong Markov property as described

in the lemma below.

In the following lemma, we consider two setups. The ﬁrst one includes extended models

with sample paths in C0(˜

G, S), coming from symmetric diﬀusions. The second one includes

Markov chains with ﬁnite state spaces, in which the extension comes from sampling Markov

chain bridges.

Lemma 2.1. Consider an extended model (Xv)v∈˜

Gassociated with a Markov process or

chain as described above and a random connected compact set K⊆˜

Gsuch that {K⊂U}is

measurable w.r.t. (X)for all open sets U. Then, conditionally on (K, X|K), the distribution

of X|Kcis described as an extended model with the same transition probabilities (i.e., any

ﬁnite marginal is described by a nearest neighbor spin system whose edge energies are of the

form (2.3) but depend on the length of the edges) with boundary condition (Xv)v∈∂K .

In the above lemma, the sigma algebra associated with X|∂K is that of ∩ε>0σ(XKε\K)

where Kεis an ε-neighborhood of K. In particular, if ∂K corresponds to the location of

a jump in the case of the extension with Markov chains, although the value at the jump is

ambiguous, the left-limit and the right-limit around it are not. This case does not occur for

extensions using a diﬀusion. An example below will be the case of the dilute Ising model,

say with some boundary conditions, for which the state space is {−1,0,1}and where Kxis

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RandomclustersintheVillainandXYmodelsJulienDubedat*HugoFalconetNovember30,2022AbstractIntheIsingandPottsmodel,randomclusterrepresentationsprovideageometricinterpretationtospincorrelations.WediscusssimilarconstructionsfortheVillainandXYmodels,wherespinstakevaluesinthecircle,aswellasextensionstomode...

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Random clusters in the Villain and XY models Julien Dub edatHugo Falconet November 30 2022

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