Convergence of the two-dimensional random walk loop soup clusters to CLE

We consider the random walk loop soup on the discrete half-plane corresponding to a central charge c in (0, 1]. We look at the clusters of discrete loops and show that the scaling limit of the outer boundaries of outermost clusters is the CLE(kappa) loop ensemble, with the same relation between kappa and c as in the continuum Brownian setting.Comment: 20 pages, 7 figure

A note on Ising random currents, Ising-FK, loop-soups and the Gaussian free field

We make a few elementary observations that relate directly the items mentioned in the title. In particular, we note that when one superimposes the random current model related to the Ising model with an independent Bernoulli percolation model with well-chosen weights, one obtains exactly the FK-percolation (or random cluster model) associated with the Ising model. We also point out that this relation can be interpreted via loop-soups, combining the description of the sign of a Gaussian Free Field on a discrete graph knowing its square (and the relation of this question with the FK-Ising model) with the loop-soup interpretation of the random current model.Comment: 5 page

An equivalence between gauge-twisted and topologically conditioned scalar Gaussian free fields

We study on the metric graphs two types of scalar Gaussian free fields (GFF), the usual one and the one twisted by a $\{-1,1\}$-valued gauge field. We show that the latter can be obtained, up to an additional deterministic transformation, by conditioning the first on a topological event. This event is that all the sign clusters of the field should be trivial for the gauge field, that is to say should not contain loops with holonomy $-1$. We also express the probability of this topological event as a ratio of two determinants of Laplacians to the power $1/2$, the usual Laplacian and the gauge-twisted Laplacian. As an example, this gives on annular planar domains the probability that no sign cluster of the metric graph GFF surrounds the inner hole of the domain.Comment: 27 pages, 7 figure

Topological expansion in Dynkin type isomorphisms for matrix valued fields

25 pages, 7 figuresWe consider Gaussian fields of symmetric or Hermitian matrices over an electrical network, and describe how Dynkin type isomorphisms with random walks for these fields make appear topological expansions encoded by ribbon graphs. A particular case of this, in continuum, is that of a Dyson's Brownian motion for β equal to 1 or 2. We further consider matrix valued Gaussian fields twisted by an orthogonal or unitary connection. In this case the isomorphisms make appear traces of holonomies of the connection along random walk loops parametrized by cycles of ribbon graphs

First passage sets of the 2D continuum Gaussian free field

We introduce the first passage set (FPS) of constant level $-a$ of the two-dimensional continuum Gaussian free field (GFF) on finitely connected domains. Informally, it is the set of points in the domain that can be connected to the boundary by a path on which the GFF does not go below $-a$. It is, thus, the two-dimensional analogue of the first hitting time of $-a$ by a one-dimensional Brownian motion. We provide an axiomatic characterization of the FPS, a continuum construction using level lines, and study its properties: it is a fractal set of zero Lebesgue measure and Minkowski dimension 2 that is coupled with the GFF $\Phi$ as a local set $A$ so that $\Phi+a$ restricted to $A$ is a positive measure. One of the highlights of this paper is identifying this measure as a Minkowski content measure in the non-integer gauge $r \mapsto \vert\log(r)\vert^{1/2}r^{2}$, by using Gaussian multiplicative chaos theory.Comment: The first version also contained arXiv:1805.09204, which is now a paper on its own; the third version is an all-around improved version ; 42 pages; 8 figures

The Vervaat transform of Brownian bridges and Brownian motion

For a continuous function $f \in \mathcal{C}([0,1])$, define the Vervaat transform $V(f)(t):=f(\tau(f)+t \mod1)+f(1)1_{\{t+\tau(f) \geq 1\}}-f(\tau(f))$, where $\tau(f)$ corresponds to the first time at which the minimum of $f$ is attained. Motivated by recent study of quantile transforms of random walks and Brownian motion, we investigate the Vervaat transform of Brownian motion and Brownian bridges with arbitrary endpoints. When the two endpoints of the bridge are not the same, the Vervaat transform is not Markovian. We describe its distribution by path decomposition and study its semi-martingale property. The same study is done for the Vervaat transform of unconditioned Brownian motion, the expectation and variance of which are also derived.Comment: 31 Pages, 4 figures. This paper (published by http://ejp.ejpecp.org/article/view/3744) is combined of two papers "On Vervaat transform of Brownian bridges and Brownian motion"(arXiv:1307.7952) by Jim Pitman and Wenpin Tang, and "Some points on Vervaat's transform of Brownian bridges and Brownian motion"(arXiv:1308.3759) by Titus Lup

Poisson ensembles of loops of one-dimensional diffusions

We study the analogue of Poisson ensembles of Markov loops ('loop soups') in the setting of one-dimensional diffusions. We give a detailed description of the corresponding intensity measure. The properties of this measure on loops lead us to an extension of Vervaat's bridge-to-excursion transformation that relates the bridges conditioned by their minimum and the excursions of all the diffusion we consider and not just the Brownian motion. Further we describe the Poisson point process of loops, their occupation fields and explain how to sample these Poisson ensembles of loops using two-dimensional Markov processes. Finally we introduce a couple of interwoven determinantal point processes on the line which is a dual through Wilson's algorithm of Poisson ensembles of loops and study the properties of these determinantal point processes.Comment: 151 page