arXiv subjects

David Vernotte

Publications and source records attributed to David Vernotte.

6 recordsLinked to original sources

arXiv2026

Shadow and percolation III: chemical distance in continuous landscapes with correlations

We study some geometric properties of the excursion set of a slope field alpha associated to a smooth, planar, centered, Gaussian field f. That, is we consider the set of all points such that the value of alpha is at most l where l is a real parameter called the level. We restrict our attention to the levels l that are supercritical. We show that for almost such l, in the sense of the Lebesgue measure, then with high probability the chemical distance between two points connected in the excursion set at level l is comparable to the usual Euclidean distance between those two points. This result is in the spirit of the Antal Pisztora theorem for Bernoulli percolation. However, many new difficulties arise such as the fact that alpha is a continuous field (not differentiable everywhere) with long range correlations and whose law is still not well understood.

math.PR↗

arXiv2025

Shadow and percolation II: discrete and continuous landscapes with correlations

In this paper we consider a discrete or continuous landscape with correlations and we consider a source of light (a sun) at infinity emitting parallel rays of light making a slope l with the horizontal plane. Depending on the value of l some portions of the landscape may be lit by the sun or be in the shadow. Under some assumptions, we show that if l is big enough then there exists a giant component of light. However, if l>0 is small enough and if we are in discrete case, then there exists a giant component of shadow. We relate this problem to the study of the percolation properties of a new random planar field.

math.PR↗

arXiv2025

Shadow and percolation I: discrete landscapes with independence

Let X be a planar random field on Z^2 which we interpret as a random height function describing some landscape of montains. We consider a source of light (a sun) located at infinity in a direction parallel with an axis od Z^2 and emitting rays which are all parallel and make a slope l with the horizontal plane. Given the value of l some montains of the landscape will be lit by the sun and other will be in the shadow of some higher mountain. Under some assumptions on X, including and independence assumption, we prove that this model may present two different phases depending on l. When l>0 is small enough then, almost surely, there exists an unbounded cluster of points in the shadow. However, if l is big enough then, almost surely, there exists an unbounded cluster of points lit by the sun. We reformulate this problem in terms of percolation of a field alpha which has a simple definition (in terms of X) but that does not present many of the nice properties usually found in percolation models such as FKG inequality, invariance by rotation or finite range correlations.

math.PR↗

arXiv2025

Chemical distance for smooth Gaussian fields in higher dimension

Gaussian percolation can be seen as the generalization of standard Bernoulli percolation on $\mathbb{Z}^d$. Instead of a random discrete configuration on a lattice, we consider a continuous Gaussian field $f$ and we study the topological and geometric properties of the random excursion set $\mathcal{E}_\ell(f) := \{x\in \mathbb{R}^d\ |\ f(x)\geq -\ell\}$ where $\ell\in \mathbb{R}$ is called a level. It is known that for a wide variety of fields $f$, there exists a phase transition at some critical level $\ell_c$. When $\ell> \ell_c$, the excursion set $\mathcal{E}_\ell(f)$ presents a unique unbounded component while if $\ell<\ell_c$ there are only bounded components in $\mathcal{E}_\ell(f)$. In the supercritical regime, $\ell>\ell_c$, we study the geometry of the unbounded cluster. Inspired by the work of Peter Antal and Agoston Pisztora for the Bernoulli model \cite{Antal}, we introduce the chemical distance between two points $x$ and $y$ as the Euclidean length of the shortest path connecting these points and staying in $\mathcal{E}_\ell(f)$. In this paper, we show that when $\ell>-\ell_c$ then with high probability, the chemical distance between two points has a behavior close to the Euclidean distance between those two points.

math.PR↗

arXiv2024

Fractal behavior for nodal lines of smooth planar Gaussian fields at criticality

This paper is devoted to the study of the large scale geometry of the excursion set and nodal set of a planar smooth Gaussian field at criticality $\ell=\ell_c=0$. We prove that there exists $s_1>1$ such that with high probability, macroscopic nodal lines in a box of size $\lambda$ are of length at least $\lambda^{s_1}$. As an application, on the event that a box is crossed by a nodal line, then the shortest crossing is of length at least $\lambda^{s_1}$. We also prove that there exists $s_2<2$ such that with high probability, the shortest crossing is non degenerated, that is, its length is at most $\lambda^{s_2}$. The argument for the lower bound is based on a celebrated paper of Aizenman and Burchard [1] that provides a general argument to show that random curves present a fractal behavior. For the upper bound, our proof relies on the polynomial decay of the probability of one-arm events which was proven in [4].

math.PR↗

arXiv2023

Chemical distance in the supercritical phase of planar Gaussian fields

Our study concerns the large scale geometry of the excursion set of planar random fields: E ${\ell}$ = {x $\in$ R 2 |f (x) $\ge$-${\ell}$}, where ${\ell}$ $\in$ R is a real parameter and f is a continuous, stationary, centered, planar Gaussian field satisfying some regularity assumptions (in particular, this study applies to the planar Bargmann-Fock field). It is already known that under those hypotheses there is a phase transition at ${\ell}$c = 0. When ${\ell}$ > 0, we are in a supercritical regime and almost surely E ${\ell}$ has a unique unbounded connected component. We prove that in this supercritical regime, whenever two points are in the same connected components of E ${\ell}$ then, with high probability, the chemical distance (the length of the shortest path in E ${\ell}$ between these points) is close to the Euclidean distance between those two points Contents

math.PR↗