35 research outputs found
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 -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 . We also express the
probability of this topological event as a ratio of two determinants of
Laplacians to the power , 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 ÎČ 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 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 . It
is, thus, the two-dimensional analogue of the first hitting time of 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 as a local set so that restricted to
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 , 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 , define the Vervaat
transform , where corresponds to the first time at which the
minimum of 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