We define a large new class of conformal primary operators in the ensemble of Brownian loops in two dimensions known as the ``Brownian loop soup,'' and compute their correlation functions analytically and in closed form. The loop soup is a conformally invariant statistical ensemble with central charge $c = 2 λ$, where $λ> 0$ is the intensity of the soup. Previous work identified exponentials of the layering operator $e^{i βN(z)}$ as primary operators. Each Brownian loop was assigned $\pm 1$ randomly, and $N(z)$ was defined to be the sum of these numbers over all loops that encircle the point $z$. These exponential operators then have conformal dimension ${\fracλ{10}}(1 - \cos β)$. Here we generalize this procedure by assigning a more general random value to each loop. The operator $e^{i βN(z)}$ remains primary with conformal dimension $\frac λ{10}(1 - ϕ(β))$, where $ϕ(β)$ is the characteristic function of the probability distribution used to assign random values to each loop. Using recent results we compute in closed form the exact two-point functions in the upper half-plane and four-point functions in the full plane of this very general class of operators. These correlation functions depend analytically on the parameters $λ, β_i, z_i$, and on the characteristic function $ϕ(β)$. They satisfy the conformal Ward identities and are crossing symmetric. As in previous work, the conformal block expansion of the four-point function reveals the existence of additional and as-yet uncharacterized conformal primary operators.
We define and study a set of operators that compute statistical properties of the Brownian Loop Soup, a conformally invariant gas of random Brownian loops (Brownian paths constrained to begin and end at the same point) in two dimensions. We prove that the correlation functions of these operators have many of the properties of conformal primaries in a conformal field theory, and compute their conformal dimension. The dimensions are real and positive, but have the novel feature that they vary continuously as a periodic function of a real parameter. We comment on the relation of the Brownian Loop…
We introduce a natural "massive" version of the Brownian loop soup of Lawler and Werner which displays conformal covariance and exponential decay. We show that this massive Brownian loop soup arises as the near-critical scaling limit of a random walk loop soup with killing and is related to the massive SLE(2) identified by Makarov and Smirnov as the near-critical scaling limit of a loop-erased random walk with killing. We conjecture that the massive Brownian loop soup describes the zero level lines of the massive Gaussian free field, and discuss possible relations to other models, such as Isin…
We compute analytically and in closed form the four-point correlation function in the plane, and the two-point correlation function in the upper half-plane, of layering vertex operators in the two dimensional conformally invariant system known as the Brownian Loop Soup. These correlation functions depend on multiple continuous parameters: the insertion points of the operators, the intensity of the soup, and the charges of the operators. In the case of the four-point function there is non-trivial dependence on five continuous parameters: the cross-ratio, the intensity, and three real charges. T…
The Brownian loop soup (BLS) is a conformally invariant statistical ensemble of random loops in two dimensions characterized by an intensity $λ>0$. Recently, we constructed families of operators in the BLS and showed that they transform as conformal primary operators. In this paper we provide an explicit expression for the BLS stress-energy tensor and compute its operator product expansion with other operators. Our results are consistent with the conformal Ward identities and our previous result that the central charge is $c = 2 λ$. In the case of domains with boundary we identify a boundary o…
We study the structure of Brownian loop-soup clusters in two dimensions. Among other things, we obtain the following decomposition of the clusters with critical intensity: When one conditions a loop-soup cluster by its outer boundary $γ$ (which is known to be an SLE(4)-type loop), then the union of all excursions away from $γ$ by all the Brownian loops in the loop-soup that touch $γ$ is distributed exactly like the union of all excursions of a Poisson point process of Brownian excursions in the domain enclosed by $γ$. A related result that we derive and use is that the couplings of the Gaussia…
We show that if one conditions a cluster in a Brownian loop-soup $L$ (of any intensity) in a two-dimensional domain by a portion $l$ of its outer boundary, then in the remaining domain, the union of all the loops of $L$ that touch $l$ satisfies the conformal restriction property while the other loops in $L$ form an independent loop-soup. This result holds when one discovers $l$ in a natural Markovian way, such as in the exploration procedures that have been defined in order to actually construct the Conformal Loop Ensembles as outer boundaries of loop-soup clusters. This result implies among o…
CEREMADE, CNRS, Université Paris-Dauphine, PSL Research University（CEREMADE，法国国家科学研究中心，巴黎第九大学，PSL研究大学）；Sorbonne Université and Université Paris Cité, CNRS, Laboratoire de Probabilités, Statistique et Modélisation（索邦大学和巴黎西岱大学，法国国家科学研究中心，概率、统计与建模实验室）
We study vertex-like operators built from the Brownian loop soup in the limit as the loop soup intensity tends to infinity. More precisely, following Camia, Gandolfi and Kleban (Nuclear Physics B 902, 2016), we take a Brownian loop soup in a planar domain and assign a random sign to each loop. We then consider random fields defined by taking, at every point of the domain, the exponential of a purely imaginary constant times the sum of the signs associated to the loops that wind around that point. As smaller loops are included in the count, that sum diverges logarithmically with the diameter of…
We study spin systems defined by the winding of a random walk loop soup. For a particular choice of loop soup intensity, we show that the corresponding spin system is reflection-positive and is dual, in the Kramers-Wannier sense, to the spin system $\text{sgn}(φ)$ where $φ$ is a discrete Gaussian free field. In general, we show that the spin correlation functions have conformally covariant scaling limits corresponding to the one-parameter family of functions studied by Camia, Gandolfi and Kleban (Nuclear Physics B, 902, 2016) and defined in terms of the winding of the Brownian loop soup. These…
The Brownian loop soup is a conformally invariant statistical ensemble of random loops in two dimensions characterized by an intensity $λ>0$, with central charge $c=2 λ$. Recent progress resulted in an analytic form for the four-point function of a class of scalar conformal primary "layering vertex operators" $\mathcal{O}_β$ with dimensions $(Δ, Δ)$, with $Δ= \fracλ{10}(1-\cosβ)$, that compute certain statistical properties of the model. The Virasoro conformal block expansion of the four-point function revealed the existence of a new set of operators with dimensions $(Δ+ k/3, Δ+ k'/3)$, for al…
The two-dimensional Brownian loop-soup is a Poissonian random collection of loops in a planar domain with an intensity parameter c. When c is not greater than 1, we show that the outer boundaries of the loop clusters are disjoint simple loops (when c>1, there is almost surely only one cluster) that satisfy certain conformal restriction axioms. We prove various results about loop-soups, cluster sizes, and the c=1 phase transition. Combining this with the results of another paper of ours on the Markovian characterization of simple conformal loop ensembles (CLE), this proves that these outer boun…
We explore the geometric meaning of the so-called zeta-regularized determinant of the Laplace-Beltrami operator on a compact surface, with or without boundary. We relate the $(-c/2)$-th power of the determinant of the Laplacian to the appropriately regularized partition function of a Brownian loop soup of intensity $c$ on the surface. This means that, in a certain sense, decorating a random surface by a Brownian loop soup of intensity $c$ corresponds to weighting the law of the surface by the $(-c/2)$-th power of the determinant of the Laplacian. Next, we introduce a method of regularizing a L…
The main topic of these lecture notes is the continuum scaling limit of planar lattice models. One reason why this topic occupies an important place in the theory of probability and mathematical statistical physics is that scaling limits provide the link between statistical mechanics and (Euclidean) field theory. In order to explain the main ideas behind the concept of scaling limit, I will focus on a "toy" model that exhibits the typical behavior of statistical mechanical models at and near the critical point. This model, known as the random walk loop soup, is actually interesting in its own …
This article deals with limit theorems for certain loop variables for loop soups whose intensity approaches infinity. We first consider random walk loop soups on finite graphs and obtain a central limit theorem when the loop variable is the sum over all loops of the integral of each loop against a given one-form on the graph. An extension of this result to the noncommutative case of loop holonomies is also discussed. As an application of the first result, we derive a central limit theorem for windings of loops around the faces of a planar graphs. More precisely, we show that the winding field …
For random collections of self-avoiding loops in two-dimensional domains, we define a simple and natural conformal restriction property that is conjecturally satisfied by the scaling limits of interfaces in models from statistical physics. This property is basically the combination of conformal invariance and the locality of the interaction in the model. Unlike the Markov property that Schramm used to characterize SLE curves (which involves conditioning on partially generated interfaces up to arbitrary stopping times), this property only involves conditioning on entire loops and thus appears a…
We introduce and compute the generalized disconnection exponents $η_κ(β)$ which depend on $κ\in(0,4]$ and another real parameter $β$, extending the Brownian disconnection exponents (corresponding to $κ=8/3$) computed by Lawler, Schramm and Werner 2001 (conjectured by Duplantier and Kwon 1988). For $κ\in(8/3,4]$, the generalized disconnection exponents have a physical interpretation in terms of planar Brownian loop-soups with intensity $c\in (0,1]$, which allows us to obtain the first prediction of the dimension of multiple points on the cluster boundaries of these loop-soups. In particular, ac…
In this research announcement, we show that SLE curves can in fact be viewed as boundaries of certain simple Poissonian percolation clusters: Recall that the Brownian loop-soup (introduced in the paper arxiv:math.PR/0304419 with Greg Lawler) with intensity c defines a Poissonian collection of (simple if one focuses only on the outer boundary) loops in a domain. This random family of (possibly intersecting) loops is conformally invariant (and there are almost surely infinitely many small loops in any sample). We show that there exists a critical value a in (0,1] such that if one colors all the …
We construct an infinite volume spatial random permutation $(\mathsf X,σ)$, where $\mathsf X\subset\mathbb R^d$ is locally finite and $σ:\mathsf X\to \mathsf X$ is a permutation, associated to the formal Hamiltonian $$ H(\mathsf X,σ) = \sum_{x\in \mathsf X} \|x-σ(x)\|^2. $$ The measures are parametrized by the point density $ρ$ and the temperature $α$. Spatial random permutations are naturally related to boson systems through a representation originally due to Feynman (1953). Let $ρ_c=ρ_c(α)$ be the critical density for Bose-Einstein condensation in Feynman's representation. Each finite cycle …
The Gaussian Free Field (GFF) in the continuum appears to be the natural generalisation of Brownian motion, when one replaces time by a multidimensional continuous parameter. The goal of these lecture notes is to describe some aspects of the continuum GFF and of its discrete counterpart defined on lattices, with the aim of providing a gentle self-contained introduction to some recent developments on this topic, such as the relation between the continuum GFF, Brownian loop-soups and the Conformal Loop Ensembles CLE(4). This is an updated and expanded version of the notes written by the first au…
Let $ρ$ be compactly supported on $D \subset \mathbb R^2$. Endow $\mathbb R^2$ with the metric $e^ρ(dx_1^2 + dx_2^2)$. As $δ\to 0$ the set of Brownian loops centered in $D$ with length at least $δ$ has measure $$\frac{\text{area}(D)}{2πδ} + \frac{1}{48π}(ρ,ρ)_{\nabla}+ o(1).$$ When $ρ$ is smooth, this follows from the classical Polyakov-Alvarez formula. We show that the above also holds if $ρ$ is not smooth, e.g. if $ρ$ is only Lipschitz. This fact can alternatively be expressed in terms of heat kernel traces, eigenvalue asymptotics, or zeta regularized determinants. Variants of this statement…
Renormalised two-point functions of CLE$_4$ gaskets
EPFL, Institute of Mathematics（洛桑联邦理工学院数学研究所）；Sorbonne Université, Université Paris Cité, CNRS, Laboratoire de Probabilités, Statistique et Modélisation, LPSM（索邦大学、巴黎西岱大学、法国国家科学研究中心概率统计与建模实验室）
The goal of the present paper is to explain, based on properties of the conformal loop ensembles CLE$_κ$ (both with simple and non-simple loops, i.e., for the whole range $κ\in (8/3, 8)$) how to derive the connection probabilities in conformal rectangles for a conditioned version of CLE$_κ$ which can be interpreted as a CLE$_κ$ with wired/free/wired/free boundary conditions on four boundary arcs (the wired parts being viewed as portions of to-be-completed loops). In particular, in the case of a conformal square, we prove that the probability that the two wired sides hook up so that they create…
