arXiv subjects

Avelio Sepúlveda

Publications and source records attributed to Avelio Sepúlveda.

29 records · Page 2Linked to original sources

arXiv2019

Dimension of two-valued sets via imaginary chaos

Two-valued sets are local sets of the two-dimensional Gaussian free field (GFF) that can be thought of as representing all points of the domain that may be connected to the boundary by a curve on which the GFF takes values only in [-a,b]. Two-valued sets exist whenever $a+b\geq 2λ$, where $λ$ depends explicitly on the normalization of the GFF. We prove that the almost sure Hausdorff dimension of the two-valued set $A_{-a,b}$ equals $d=2-2λ^2/(a+b)^2$. For the two-point estimate, we use the real part of a "vertex field" built from the purely imaginary Gaussian multiplicative chaos. We also construct a non-trivial $d$-dimensional measure supported on $A_{-a,b}$ and discuss its relation with the $d$-dimensional conformal Minkowski content for $A_{-a,b}$.

math.PR↗

arXiv2019

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 $Φ$ as a local set $A$ so that $Φ+a$ 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.

math.PR↗

arXiv2019

Liouville dynamical percolation

We construct and analyze a continuum dynamical percolation process which evolves in a random environment given by a $γ$-Liouville measure. The homogeneous counterpart of this process describes the scaling limit of discrete dynamical percolation on the rescaled triangular lattice. Our focus here is to study the same limiting dynamics, but where the speed of microscopic updates is highly inhomogeneous in space and is driven by the $γ$-Liouville measure associated with a two-dimensional log-correlated field $h$. Roughly speaking, this continuum percolation process evolves very rapidly where the field $h$ is high and barely moves where the field $h$ is low. Our main results can be summarized as follows. 1. First, we build this inhomogeneous dynamical percolation which we call $γ$-Liouville dynamical percolation (LDP) by taking the scaling limit of the associated process on the triangular lattice. We work with three different regimes each requiring different tools: $γ\in [0,2-\sqrt{5/2})$, $γ\in [2-\sqrt{5/2}, \sqrt{3/2})$, and $γ\in(\sqrt{3/2},2)$. 2. When $γ<\sqrt{3/2}$, we prove that $γ$-LDP is mixing in the Schramm-Smirnov space as $t\to \infty$, quenched in the log-correlated field $h$. On the contrary, when $γ>\sqrt{3/2}$ the process is frozen in time. The ergodicity result is a crucial piece of the Cardy embedding project of the second and fourth coauthors, where LDP for $γ=\sqrt{1/6}$ is used to study the scaling limit of a variant of dynamical percolation on uniform triangulations. 3. When $γ<\sqrt{3/4}$, we obtain quantitative bounds on the mixing of quad crossing events.

math.PR↗

arXiv2019

Tree-decorated planar maps

We introduce the set of (non-spanning) tree-decorated planar maps, and show that they are in bijection with the Cartesian product between the set of trees and the set of maps with a simple boundary. As a consequence, we count the number of tree decorated triangulations and quadrangulations with a given amount of faces and for a given size of the tree. Finally, we generalise the bijection to study other types of decorated planar maps and obtain explicit counting formulas for them.

math.CO↗

arXiv2019

Critical Liouville measure as a limit of subcritical measures

We study how the Gaussian multiplicative chaos (GMC) measures $μ^γ$ corresponding to the 2D Gaussian free field change when $γ$ approaches the critical parameter $2$. In particular, we show that as $γ\to 2^{-}$, $(2-γ)^{-1}μ^γ$ converges in probability to $2μ'$, where $μ'$ is the critical GMC measure.

math.PR↗

arXiv2018

Negative moments for Gaussian multiplicative chaos on fractal sets

The objective of this note is to study the probability that the total mass of a sub-critical Gaussian multiplicative chaos (GMC) with arbitrary base measure $σ$ is small. When $σ$ has some continuous density w.r.t Lebesgue measure, a scaling argument shows that the logarithm of the total GMC mass is sub-Gaussian near $-\infty$. However, when $σ$ has no scaling properties, the situation is much less clear. In this paper, we prove that for any base measure $σ$, the total GMC mass has negative moments of all orders.

math.PR↗

arXiv2018

On thin local sets of the Gaussian free field

We study how small a local set of the continuum Gaussian free field (GFF) in dimension $d$ has to be to ensure that this set is thin, which loosely speaking means that it captures no GFF mass on itself, in other words, that the field restricted to it is zero. We provide a criterion on the size of the local set for this to happen, and on the other hand, we show that this criterion is sharp by constructing small local sets that are not thin.

math.PR↗

arXiv2018

The first passage sets of the 2D Gaussian free field: convergence and isomorphisms

In a previous article, we introduced 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 along which the GFF is greater than or equal to $-a$. This description can be taken as a definition of the FPS for the metric graph GFF, and it justifies the analogy with the first hitting time of $-a$ by a one-dimensional Brownian motion. In the current article, we prove that the metric graph FPS converges towards the continuum FPS in the Hausdorff metric. This allows us to show that the FPS of the continuum GFF can be represented as a union of clusters of Brownian excursions and Brownian loops, and to prove that Brownian loop soup clusters admit a non-trivial Minkowski content in the gauge $r\mapsto |\log r|^{1/2}r^2$. We also show that certain natural interfaces of the metric graph GFF converge to SLE$_4$ processes.

math.PR↗

arXiv2018

Two-valued local sets of the 2D continuum Gaussian free field: connectivity, labels, and induced metrics

We study two-valued local sets, $\mathbb{A}_{-a,b}$, of the two-dimensional continuum Gaussian free field (GFF) with zero boundary condition in simply connected domains. Intuitively, $\mathbb{A}_{-a,b}$ is the (random) set of points connected to the boundary by a path on which the values of the GFF remain in $[-a,b]$. For specific choices of the parameters $a, b$ the two-valued sets have the law of the CLE$_4$ carpet, the law of the union of level lines between all pairs of boundary points, or, conjecturally, the law of the interfaces of the scaling limit of XOR-Ising model. Two-valued sets are the closure of the union of countably many SLE$_4$ type of loops, where each loop comes with a label equal to either $-a$ or $b$. One of the main results of this paper describes the connectivity properties of these loops. Roughly, we show that all the loops are disjoint if $a+b \geq 4λ$, and that their intersection graph is connected if $a + b < 4λ$. This also allows us to study the labels (the heights) of the loops. We prove that the labels of the loops are a function of the set $\mathbb{A}_{-a,b}$ if and only if $a\neq b$ and $2λ\leq a+b < 4λ$ and that the labels are independent given the set if and only if $a = b = 2λ$. We also show that the threshold for the level-set percolation in the 2D continuum GFF is $-2λ$. Finally, we discuss the coupling of the labelled CLE$_4$ with the GFF. We characterise this coupling as a specific local set coupling, and show how to approximate these local sets. We further see how in these approximations the labels naturally encode distances to the boundary.

math.PR↗

arXiv2017

Liouville measure as a multiplicative cascade via level sets of the Gaussian free field

We provide new constructions of the subcritical and critical Gaussian multiplicative chaos (GMC) measures corresponding to the 2D Gaussian free field (GFF). As a special case we recover E. Aidekon's construction of random measures using nested conformally invariant loop ensembles, and thereby prove his conjecture that certain CLE$_4$ based limiting measures are equal in law to the GMC measures for the GFF. The constructions are based on the theory of local sets of the GFF and build a strong link between multiplicative cascades and GMC measures. This link allows us to directly adapt techniques used for multiplicative cascades to the study of GMC measures of the GFF. As a proof of principle we do this for the so-called Seneta--Heyde rescaling of the critical GMC measure.

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↗