arXiv subjects

Nicolas Bouchot

Publications and source records attributed to Nicolas Bouchot.

6 recordsLinked to original sources

arXiv2025

Covering of an inner subset by the confined random walk

We consider the simple random walk conditioned to stay forever in a finite domain $D_N \subset \mathbb{Z}^d, d \geq 3$ of typical size $N$. This confined walk is a random walk on the conductances given by the first eigenvector of the Laplacian on $D_N$. On inner sets of $D_N$, the trace of this confined walk can be approximated by tilted random interlacements, which is a useful tool to understand some properties of the walk. In this paper, we propose to study the cover time of inner subsets $\Lambda_N$ of $D_N$ as well as the so-called late points of these subsets. If $\Lambda_N$ contains enough late points, we obtain the asymptotic expansion of the covering time as $c_\Lambda N^d \big[ \log N - \log\log N + \mathcal{G} \big]$, with $\mathcal{G}$ a Gumbel random variable, as well as a Poisson repartition of these late points. The method we use is similar to Belius' work about the simple random walk on the torus, which displays the same asymptotics albeit without the $\log \log N$ term. In the more general setting of ``ball-like'' $\Lambda_N$, we simply get the first term of the asymptotic expansion.

math.PR↗

arXiv2025

How thin does random interlacement have to be so that a random walk can see through it?

The random interlacements $\mathscr{I}(u)$ at level $u$ has been introduced by Sznitman, as a Poissonian collection of independent simple random walk trajectories on $\mathbb{Z}^d$, $d\geq 3$, with intensity $u>0$. Since then, several works investigated the properties of the random interlacements intersected with large sets of~$\mathbb{Z}^d$. In this paper, we study the asymptotic behavior of the capacity of $\mathscr{I}(u) \cap D_N$, where $D_N$ is the blow up of a compact set $D$, with typical size $N$. We determine the correct window $(u_N)_{N\geq 1}$ of the intensity parameter for which the capacity $\mathrm{cap}(\mathscr{I}(u_N)\cap D_N)$ starts to become negligible compared to $\mathrm{cap}(D_N)$; this roughly means that a random walk starting from far away starts to see through $\mathscr{I}(u_N)\cap D_N$. In the same spirit, we investigate the capacity of the simple random walk conditioned to stay in a large Euclidean ball up to time $t_N$, and find similar asymptotics by taking $t_N = u_N N^d$.

math.PR↗

arXiv2024

Some properties of the principal Dirichlet eigenfunction in Lipschitz domains, via probabilistic couplings

We study a discrete and continuous version of the spectral Dirichlet problem in an open bounded connected set $\Omega\subset \mathbb{R}^d$, in dimension $d\geq 2$. More precisely, consider the simple random walk on $\mathbb{Z}^d$ killed upon exiting the (large) bounded domain $\Omega_N = (N\Omega)\cap \mathbb{Z}^d$. We let $P_N$ its transition matrix and we study the properties of its ($L^2$-normalized) principal eigenvector $\phi_N$, also known as ground state. Under mild assumptions on $\Omega$, we give regularity estimates on $\phi_N$, namely on its $k$-th order differences (or $k$-th order derivatives), with a uniform control inside $\Omega_N$. We provide a completely probabilistic proof of these estimates: our starting point is a Feynman-Kac representation of $\phi_N$, combined with gambler's ruin estimates and a new ``multi-mirror'' coupling, which may be of independent interest. We also obtain the same type of estimates for the first eigenfunction $\varphi_1$ of the corresponding continuous spectral Dirichlet problem, in relation with a Brownian motion killed upon exiting $\Omega$. Finally, we take the opportunity to review (and slightly extend) some of the literature on the $L^2$ and uniform convergence of $\phi_N$ to $\varphi_1$ in Lipschitz bounded domains of $\mathbb{R}^d$, which can be derived thanks to our estimates.

math.PR↗

arXiv2024

A confined random walk locally looks like tilted random interlacements

In this paper we consider the simple random walk on $\mathbb{Z}^d$, $d \geq 3$, conditioned to stay in a large domain $D_N$ of typical diameter $N$. Considering the range up to time $t_N \geq N^{2+\delta}$ for some $\delta > 0$, we establish a coupling with what Teixeira (2009) and Li & Sznitman (2014) defined as "tilted random interlacements". This tilted interlacement can be described as random interlacements but with trajectories given by random walks on conductances $c_N(x,y) = \phi_N(x) \phi_N(y)$, where $\phi_N$ is the first eigenvector of the discrete Laplace-Beltrami operator on $D_N$. The coupling follows the methodology of the soft local times, introduced by Popov & Teixeira (2015) and used by \v{C}ern\'y & Teixeira (2016) to prove the well-known coupling between the simple random walk on the torus and the random interlacements.

math.PR↗

arXiv2023

Scaling limit of a one-dimensional polymer in a repulsive i.i.d. environment

The purpose of this paper is to study a one-dimensional polymer penalized by its range and placed in a random environment $\omega$. The law of the simple symmetric random walk up to time $n$ is modified by the exponential of the sum of $\beta \omega_z - h$ sitting on its range, with~$h$ and $\beta$ positive parameters. It is known that, at first order, the polymer folds itself to a segment of optimal size $c_h n^{1/3}$ with $c_h = \pi^{2/3} h^{-1/3}$. Here we study how disorder influences finer quantities. If the random variables $\omega_z$ are i.i.d.\ with a finite second moment, we prove that the left-most point of the range is located near $-u_* n^{1/3}$, where $u_* \in [0,c_h]$ is a constant that only depends on the disorder. This contrast with the homogeneous model (i.e. when $\beta=0$), where the left-most point has a random location between $-c_h n^{1/3}$ and $0$. With an additional moment assumption, we are able to show that the left-most point of the range is at distance $\mathcal U n^{2/9}$ from $-u_* n^{1/3}$ and the right-most point at distance $\mathcal V n^{2/9}$ from $(c_h-u_*) n^{1/3}$. Here again, $\mathcal{U}$ and $\mathcal{V}$ are constants that depend only on $\omega$.

math.PR↗

arXiv2022

Scaling limits for the random walk penalized by its range in dimension one

In this article we study a one dimensional model for a polymer in a poor solvent: the random walk on $\mathbb{Z}$ penalized by its range. More precisely, we consider a Gibbs transformation of the law of the simple symmmetric random walk by a weight $\exp(-h_n|R_n|)$, with $|R_n|$ the number of visited sites and $h_n$ a size-dependent positive parameter. We use gambler's ruin estimates to obtain exact asymptotics for the partition function, that enables us to obtain a precise description of trajectories, in particular scaling limits for the center and the amplitude of the range. A phase transition for the fluctuations around an optimal amplitude is identified at $h_n \approx n^{1/4}$ , inherent to the underlying lattice structure.

math.PR↗