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Wendelin Werner

Publications and source records attributed to Wendelin Werner.

At least 19 recordsLinked to original sources
arXiv2026

Intensity doubling for Brownian loop-soups in high dimensions

We derive an intensity doubling feature of critical Brownian loop-soups on the cable-graphs of ${\mathbb Z}^d$ for $d \ge 7$ that can be described as follows: In the box $[-N, N]^d$ (and with a probability that goes to $1$ as $N$ goes to infinity), the set of all clusters of Brownian loops that do contain proper self-avoiding cycles of diameter comparable to $N$ can be decomposed into two identically distributed families: (a) The collection of clusters that do contain a large Brownian loop from the loop-soup (and therefore do automatically contain such a large cycle) (b) The collection of clusters that contain no macroscopic loop from the loop-soup (more specifically, no loop of diameter greater than $N^β$ when $β> 4/ (d-2)$ is fixed) but nevertheless contain a large cycle. In particular, due to the fact that these two families are asymptotically identically distributed, large cycles formed in case (b) by chains of small Brownian loops (i.e., all of diameter much smaller than $N$) will look like large Brownian loops themselves, and form a second independent "ghost" critical loop-soup in the scaling limit. Reformulated in terms of the Gaussian free field on such cable-graphs, this shows that large cycles in the collection of its sign clusters will converge in the scaling limit to a Brownian loop-soup with twice the usual critical intensity. This result had been conjectured by the first author in arXiv:2209.07901 [math.PR] ; our proof builds heavily on the second author's switching property for such loop-soups from arXiv:2502.06754 [math.PR] .

math.PR↗
arXiv2025

Parity questions in critical planar Brownian loop-soups (or "where did the free planar bosons go?")

The critical two-dimensional Brownian loop-soup is an infinite collection of non-interacting Brownian loops in a planar domain that possesses some combinatorial features related to the notion of indistinguishability of bosons. The properly renormalized occupation time field of this collection of loops is known to be distributed like the properly defined square of a Gaussian free field. In the present paper, we investigate aspects of the question about how much information these fields provide about the loop-soup. Among other things, we show that the exact set of points that are actually visited by some loops in the loop-soup is not determined by these fields. We further prove that given the fields, a dense family of special points will each have a conditional probability 1/2 of being part of the loop-soup. We also exhibit another instance where the possible decompositions (given the field) into individual loops and excursions can be grouped into two clearly different groups, each having a conditional probability 1/2 of occurring.

math.PR↗
arXiv2025

A switching identity for cable-graph loop soups and Gaussian free fields

We derive a "switching identity" that can be stated for critical Brownian loop-soups or for the Gaussian free field on a cable graph: It basically says that at the level of cluster configurations and at the more general level of the occupation time fields, conditioning two points on the cable-graph to belong to the same cluster of Brownian loops (or equivalently to the same sign-cluster of the GFF) amounts to adding a random odd number of independent Brownian excursions between these points to an otherwise unconditioned configuration. This explicit simple description of the conditional law of the clusters when a connection occurs has various direct consequences, in particular about the large scale behaviour of these sign-clusters on infinite graphs.

math.PR↗
arXiv2025

On Loops in critical high-dimensional percolation

We show the following results about critical Bernoulli percolation in high dimensions: In a box of side-length N, there exist self-avoiding open loops of diameter comparable to N, and the collection of these self-avoiding loops has a non-trivial scaling limit (if viewed in the Hausdorff topology) as N tends to infinity. This feature contrasts with the proliferation of "typical" percolation clusters pointed out by Michael Aizenman almost three decades ago. In other words, we show that among the many large clusters in a large box, only a handful will contain a self-avoiding loop of diameter greater than a fixed fraction of the side-length of the box.

math.PR↗
arXiv2021

Simple Conformal Loop Ensembles on Liouville Quantum Gravity

We show that when one draws a simple conformal loop ensemble (CLE$_κ$ for $κ\in (8/3,4)$) on an independent $\sqrtκ$-Liouville quantum gravity (LQG) surface and explores the CLE in a natural Markovian way, the quantum surfaces (e.g., corresponding to the interior of the CLE loops) that are cut out form a Poisson point process of quantum disks. This construction allows us to make direct links between CLE on LQG, asymmetric $(4/κ)$-stable processes, and labeled branching trees. The ratio between positive and negative jump intensities of these processes turns out to be $-\cos (4 π/ κ)$, which can be interpreted as a "density" of CLE loops in the CLE on LQG setting. Positive jumps correspond to the discovery of a CLE loop (where the LQG length of the loop is given by the jump size) and negative jumps correspond to the moments where the discovery process splits the remaining to be discovered domain into two pieces. Some consequences are the following: (i) It provides a construction of a CLE on LQG as a patchwork/welding of quantum disks. (ii) It allows to construct the "natural quantum measure" that lives in a CLE carpet. (iii) It enables us to derive some new properties and formulas for SLE processes and CLE themselves (without LQG) such as the exact distribution of the trunk of the general asymmetric SLE$_κ(κ-6)$ processes. The present work deals directly with structures in the continuum and makes no reference to discrete models, but our calculations match those for scaling limits of O(N) models on planar maps with large faces and CLE on LQG. Indeed, our Lévy-tree descriptions are exactly the ones that appear in the study of the large-scale limit of peeling of discrete decorated planar maps such as in recent work of Bertoin, Budd, Curien and Kortchemski. The case of non-simple CLEs on LQG is the topic of another paper.

math.PR↗
arXiv2021

Non-simple conformal loop ensembles on Liouville quantum gravity and the law of CLE percolation interfaces

We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble CLE$_{κ'}$ for $κ'$ in $(4,8)$ that is drawn on an independent $γ$-LQG surface for $γ^2=16/κ'$. The results are similar in flavor to the ones from our paper dealing with CLE$_κ$ for $κ$ in $(8/3,4)$, where the loops of the CLE are disjoint and simple. In particular, we encode the combined structure of the LQG surface and the CLE$_{κ'}$ in terms of stable growth-fragmentation trees or their variants, which also appear in the asymptotic study of peeling processes on decorated planar maps. This has consequences for questions that do a priori not involve LQG surfaces: Our previous paper "CLE percolations" described the law of interfaces obtained when coloring the loops of a CLE$_{κ'}$ independently into two colors with respective probabilities $p$ and $1-p$. This description was complete up to one missing parameter $ρ$. The results of the present paper about CLE on LQG allow us to determine its value in terms of $p$ and $κ'$. It shows in particular that CLE$_{κ'}$ and CLE$_{16/κ'}$ are related via a continuum analog of the Edwards-Sokal coupling between FK$_q$ percolation and the $q$-state Potts model (which makes sense even for non-integer $q$ between $1$ and $4$) if and only if $q=4\cos^2(4π/κ')$. This provides further evidence for the long-standing belief that CLE$_{κ'}$ and CLE$_{16/κ'}$ represent the scaling limits of FK$_q$ percolation and the $q$-Potts model when $q$ and $κ'$ are related in this way. Another consequence of the formula for $ρ(p,κ')$ is the value of half-plane arm exponents for such divide-and-color models (a.k.a. fuzzy Potts models) that turn out to take a somewhat different form than the usual critical exponents for two-dimensional models.

math.PR↗
arXiv2021

Lecture notes on the Gaussian Free Field

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 author for graduate courses at ETH Zürich. The exercises that are interspersed in the first half of these notes mostly originate from the exercise sheets prepared by the second author for this course in 2018.

math.PR↗
arXiv2020

On clusters of Brownian loops in d dimensions

We discuss random geometric structures obtained by percolation of Brownian loops, in relation to the Gaussian Free Field, and how their existence and properties depend on the dimension of the ambient space. We formulate a number of conjectures for the cases d=3,4,5 and prove some results when d > 6.

math.PR↗
arXiv2019

Near-critical spanning forests and renormalization

We study random two-dimensional spanning forests in the plane that can be viewed both in the discrete case and in their appropriately taken scaling limits as a uniformly chosen spanning tree with some Poissonian deletion of edges or points. We show how to relate these scaling limits to a stationary distribution of a natural coalescent-type Markov process on a state-space of abstract graphs with real-valued edge-weights. This Markov process can be interpreted as a renormalization flow. This provides a model for which one can rigorously implement the formalism proposed by the third author in order to relate the law of the scaling limit of a critical model to a stationary distribution of such a renormalization/Markov process: When starting from any two-dimensional lattice with constant edge-weights, the Markov process does indeed converge in law to this stationary distribution that corresponds to a scaling limit of UST with Poissonian deletions. The results of this paper heavily build on the convergence in distribution of branches of the UST to SLE$_2$ (a result by Lawler, Schramm and Werner) as well as on the convergence of the suitably renormalized length of the loop-erased random walk to the "natural parametrization" of the SLE$_2$ (a recent result by Lawler and Viklund).

math.PR↗
arXiv2018

Non-simple SLE curves are not determined by their range

We show that when observing the range of a chordal SLE$_κ$ curve for $κ\in (4,8)$, it is not possible to recover the order in which the points have been visited. We also derive related results about conformal loop ensembles (CLE): (i) The loops in a CLE$_κ$ for $κ\in (4,8)$ are not determined by the CLE$_κ$ gasket. (ii) The continuum percolation interfaces defined in the fractal carpets of conformal loop ensembles CLE$_κ$ for $κ\in (8/3, 4)$ (we defined these percolation interfaces in previous work, and showed there that they are SLE$_{16/κ}$ curves) are not determined by the CLE$_κ$ carpet that they are defined in.

math.PR↗
arXiv2018

Connection probabilities for conformal loop ensembles

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 one single loop is equal to $1/(1 - 2 \cos (4 π/ κ))$. Comparing this with the corresponding connection probabilities for discrete O(N) models for instance indicates that if a dilute O(N) model (respectively a critical FK(q)-percolation model on the square lattice) has a non-trivial conformally invariant scaling limit, then necessarily this scaling limit is CLE$_κ$ where $κ$ is the value in $(8/3, 4]$ such that $-2 \cos (4 π/ κ)$ is equal to $N$ (resp. the value in $[4,8)$ such that $-2 \cos (4π/ κ)$ is equal to $\sqrt {q}$). Our arguments and computations build on the one hand on Dubédat's SLE commutation relations (as developed and used by Dubédat, Zhan or Bauer-Bernard-Kytölä) and on the other hand, on the construction and properties of the conformal loop ensembles and their relation to Brownian loop-soups, restriction measures, and the Gaussian free field, as recently derived in works with Sheffield and with Qian.

math.PR↗
arXiv2018

Coupling the Gaussian free fields with free and with zero boundary conditions via common level lines

We point out a new simple way to couple the Gaussian Free Field (GFF) with free boundary conditions in a two-dimensional domain with the GFF with zero boundary conditions in the same domain: Starting from the latter, one just has to sample at random all the signs of the height gaps on its boundary-touching zero-level lines (these signs are alternating for the zero-boundary GFF) in order to obtain a free boundary GFF. Constructions and couplings of the free boundary GFF and its level lines via soups of reflected Brownian loops and their clusters are also discussed. Such considerations show for instance that in a domain with an axis of symmetry, if one looks at the overlay of a single usual Conformal Loop Ensemble CLE(3) with its own symmetric image, one obtains the CLE(4)-type collection of level lines of a GFF with mixed zero/free boundary conditions in the half-domain.

math.PR↗
arXiv2017

CLE percolations

Conformal loop ensembles are random collections of loops in a simply connected domain, whose laws are characterized by a natural conformal invariance property. The set of points not surrounded by any CLE loop is a natural random and conformally invariant analog of the Sierpinski gasket or carpet. In the present paper, we derive a direct relationship between each CLE consisting of simple disjoint loops (CLE($κ$) with $κ$ between 8/3 and 4) and the corresponding CLE($κ'$) where $κ':=16/κ$, a CLE consisting of non-disjoint loops. This is the continuum analog of the Edwards-Sokal coupling (between the q-state Potts model and the associated FK random cluster model) and its generalization to non-integer q. Like its discrete analog, our continuum correspondence has two directions. First, we show that one can construct (variants of) CLE($κ$) as follows: sample a CLE($κ'$), then use a biased coin to independently color each loop one of two colors, and then consider the outer boundaries of the clusters of loops of a given color. Second, we show how to interpret CLE($κ'$) loops as interfaces of a continuum analog of critical percolation within a CLE($κ)$ carpet. This is the first description of continuous percolation interfaces in fractal domains. These constructions provide new interpretations of the relationship between CLEs and the Gaussian free field. Along the way, we obtain results about generalized SLE$(κ;ρ)$ curves, and define a continuous family of natural CLE variants called boundary conformal loop ensembles (BCLEs) that share some (but not all) of the conformal symmetries that characterize CLEs, and that should be scaling limits of critical models with special boundary conditions. We extend the CLE correspondence to a BCLE correspondence that makes sense for all $κ$ between 2 and 4.

math.PR↗
arXiv2017

Decomposition of Brownian loop-soup clusters

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 Gaussian Free Field (GFF) with CLE(4) via level-lines (by Miller-Sheffield), of the square of the GFF with loop-soups via occupation times (by Le Jan), and of the CLE(4) with loop-soups via loop-soup clusters (by Sheffield and Werner) can be made to coincide. An instrumental role in our proof of this fact is played by Lupu's description of CLE(4) as limits of discrete loop-soup clusters.

math.PR↗
arXiv2017

The random pseudo-metric on a graph defined via the zero-set of the Gaussian free field on its metric graph

We further investigate properties of the Gaussian free field (GFF) on the metric graph associated to a discrete weighted graph (where the edges of the latter are replaced by continuous line-segments of appropriate length) that has been introduced by the first author. On such a metric graph, the GFF is a random continuous function that generalises one-dimensional Brownian bridges so that one-dimensional techniques can be used. In the present paper, we define and study the pseudo-metric defined on the metric graph (and therefore also on the discrete graph itself), where the length of a path on the metric graph is defined to be the local time at level zero accumulated by the Gaussian free field along this path. We first derive a pathwise transformation that relates the GFF on the metric graph with the reflected GFF on the metric graph via the pseudo-distance defined by the latter. This is a generalisation of Paul Lévy's result relating the local time at zero of Brownian motion to the supremum of another Brownian motion. We also compute explicitly the distribution of certain functionals of this pseudo-metric and of the GFF. In particular, we point out that when the boundary consists of just two points, the law of the pseudo-distance between them depends solely on the resistance of the network between them. We then discuss questions related to the scaling limit of this pseudo-metric in the two-dimensional case, which should be the conformally invariant way to measure distances between CLE(4) loops introduced and studied by the second author with Wu, and by Sheffield, Watson and Wu. Our explicit laws on metric graphs also lead to new conjectures for related functionals of the continuum GFF on fairly general Riemann surfaces.

math.PR↗
arXiv2017

On bounded-type thin local sets of the two-dimensional Gaussian free field

We study certain classes of local sets of the two-dimensional Gaussian free field (GFF) in a simply-connected domain, and their relation to the conformal loop ensemble CLE(4) and its variants. More specifically, we consider bounded-type thin local sets (BTLS), where thin means that the local set is small in size, and bounded-type means that the harmonic function describing the mean value of the field away from the local set is bounded by some deterministic constant. We show that a local set is a BTLS if and only if it is contained in some nested version of the CLE(4) carpet, and prove that all BTLS are necessarily connected to the boundary of the domain. We also construct all possible BTLS for which the corresponding harmonic function takes only two prescribed values and show that all these sets (and this includes the case of CLE(4)) are in fact measurable functions of the GFF.

math.PR↗
arXiv2015

On the spatial Markov property of soups of unoriented and oriented loops

We describe simple properties of some soups of unoriented Markov loops and of some soups of oriented Markov loops that can be interpreted as a spatial Markov property of these loop-soups. This property of the latter soup is related to well-known features of the uniform spanning trees (such as Wilson's algorithm) while the Markov property of the former soup is related to the Gaussian Free Field and to identities used in the foundational papers of Symanzik, Nelson, and of Brydges, Fröhlich and Spencer or Dynkin, or more recently by Le Jan.

math.PR↗