arXiv subjects

Guillaume Conchon--Kerjan

Publications and source records attributed to Guillaume Conchon--Kerjan.

7 recordsLinked to original sources

arXiv2026

Law of Large Numbers for a random walk on dynamic environments with drift

We study a random walk driven by a particle system from a generic class, and establish a law of large numbers for the walk for almost all densities of the environment. To do so, we exploit the finite-ranged approximations of the environment from arXiv:2409.02096 in a new way, whereby the monotonicity (in the density) of the walker's displacement is leveraged to show the existence of an actual speed. This bypasses the constructions in arXiv:1906.03167 and generalises its Theorem 1.1, which applied to specific environments. We illustrate this with a family of particle systems where the particles have underlying drifts, namely mixtures of APCRWs (Asymmetric Poisson Cloud of Random Walks). In particular, when all particles have the same drift, we prove the LLN under any choice of parameters save one critical density of the environment. To our knowledge, this is the first time that such a conservative and slow-mixing environment with drift is treated outside of the non-nestling case (in which the walker is already assumed to travel strictly faster/slower than the drift arXiv:2205.00282).

math.PR↗

arXiv2024

Sharp threshold for the ballisticity of the random walk on the exclusion process

We study a non-reversible random walk advected by the symmetric simple exclusion process, so that the walk has a local drift of opposite sign when sitting atop an occupied or an empty site. We prove that the back-tracking probability of the walk exhibits a sharp transition as the density $ρ$ of particles in the underlying exclusion process varies across a critical density $ρ_c$. Our results imply that the speed $v=v(ρ)$ of the walk is a strictly monotone function and that the zero-speed regime is either absent or collapses to a single point, $ρ_c$, thus solving a conjecture of arXiv:1906.03167. The proof proceeds by exhibiting a quantitative monotonicity result for the speed of a truncated model, in which the environment is renewed after a finite time horizon $L$. The truncation parameter $L$ is subsequently pitted against the density $ρ$ to carry estimates over to the full model. Our strategy is somewhat reminiscent of certain techniques recently used to prove sharpness results in percolation problems. A key instrument is a combination of renormalisation arguments with refined couplings of environments at slightly different densities, which we develop in this article. Our results hold in fact in greater generality and apply to a class of environments with possibly egregious features, outside perturbative regimes.

math.PR↗

arXiv2023

Speed of the random walk on the supercritical Gaussian Free Field percolation on regular trees

In this paper, we study the random walk on a supercritical branching process with an uncountable and unbounded set of types supported on the $d$-regular tree $\mathbb{T}_d$ ($d\geq 3$), namely the cluster $\mathcal{C}_\circ^h$ of the root in the level set of the Gaussian Free Field (GFF) above an arbitrary value $h\in (-\infty, h_{\star})$. The value $h_{\star}\in (0,\infty)$ is the percolation threshold; in particular, $\mathcal{C}_\circ^h$ is infinite with positive probability. We show that on $\mathcal{C}_\circ^h$ conditioned to be infinite, the simple random walk is ballistic, and we give a law of large numbers and a Donsker theorem for its speed. To do so, we design a renewal construction that withstands the long-range dependencies in the structure of the tree. This allows us to translate underlying ergodic properties of $\mathcal{C}_\circ^h$ into regularity estimates for the random walk.

math.PR↗

arXiv2023

Anatomy of a gaussian giant: supercritical level-sets of the free field on random regular graphs

In this paper, we study the level-set of the zero-average Gaussian Free Field on a uniform random $d$-regular graph above an arbitrary level $h\in (-\infty, h_{\star})$, where $h_{\star}$ is the level-set percolation threshold of the GFF on the $d$-regular tree $\mathbb{T}_d$. We prove that w.h.p as the number $n$ of vertices diverges, the GFF has a unique giant connected component $\mathcal{C}_1^{(n)}$ of size $η(h) n+o(n)$, where $η(h)$ is the probability that the root percolates in the corresponding GFF level-set on $\mathbb{T}_d$. This gives a positive answer to the conjecture of \cite{ACregulgraphs} for most regular graphs. We also prove that the second largest component has size $Θ(\log n)$. Moreover, we show that $\mathcal{C}_1^{(n)}$ shares the following similarities with the giant component of the supercritical Erdős-Rényi random graph. First, the diameter and the typical distance between vertices are $Θ(\log n)$. Second, the $2$-core and the kernel encompass a given positive proportion of the vertices. Third, the local structure is a branching process conditioned to survive, namely the level-set percolation cluster of the root in $\mathbb{T}_d$ (in the Erdős-Rényi case, it is known to be a Galton-Watson tree with a Poisson distribution for the offspring).

math.PR↗

arXiv2023

Scaling limit of critical random trees in random environment

We consider Bienaymé-Galton-Watson trees in random environment, where each generation $k$ is attributed a random offspring distribution $μ_k$, and $(μ_k)_{k\geq 0}$ is a sequence of independent and identically distributed random probability measures. We work in the ``strictly critical'' regime where, for all $k$, the average of $μ_k$ is assumed to be equal to $1$ almost surely, and the variance of $μ_k$ has finite expectation. We prove that, for almost all realizations of the environment (more precisely, under some deterministic conditions that the random environment satisfies almost surely), the scaling limit of the tree in that environment, conditioned to be large, is the Brownian continuum random tree. The habitual techniques used for standard Bienaymé-Galton-Watson trees, or trees with exchangeable vertices, do not apply to this case. Our proof therefore provides alternative tools.

math.PR↗

arXiv2021

The stable graph: the metric space scaling limit of a critical random graph with i.i.d. power-law degrees

We prove a metric space scaling limit for a critical random graph with independent and identically distributed degrees having power-law tail behaviour with exponent $α+1$, where $α\in (1,2)$. The limiting components are constructed from random $\mathbb{R}$-trees encoded by the excursions above its running infimum of a process whose law is locally absolutely continuous with respect to that of a spectrally positive $α$-stable Lévy process. These spanning $\mathbb{R}$-trees are measure-changed $α$-stable trees. In each such $\mathbb{R}$-tree, we make a random number of vertex-identifications, whose locations are determined by an auxiliary Poisson process. This generalises results which were already known in the case where the degree distribution has a finite third moment (a model which lies in the same universality class as the Erdős--Rényi random graph) and where the role of the $α$-stable Lévy process is played by a Brownian motion.

math.PR↗

arXiv2019

Cutoff for random lifts of weighted graphs

We prove a cutoff for the random walk on random $n$-lifts of finite weighted graphs, even when the random walk on the base graph $\mathcal{G}$ of the lift is not reversible. The mixing time is w.h.p. $t_{mix}=h^{-1}\log n$, where $h$ is a constant associated to $\mathcal{G}$, namely the entropy of its universal cover. Moreover, this mixing time is the smallest possible among all $n$-lifts of $\mathcal{G}$. In the particular case where the base graph is a vertex with $d/2$ loops, $d$ even, we obtain a cutoff for a $d$-regular random graph (as did Lubetzky and Sly in \cite{cutoffregular} with a slightly different distribution on $d$-regular graphs, but the mixing time is the same).

math.PR↗