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Marie Théret

Publications and source records attributed to Marie Théret.

At least 19 recordsLinked to original sources
arXiv2026

The $K$-th nearest neighbor random walk on a Poisson point process gets trapped

The $K$-th nearest neighbor random walk $(X_n)_{n \geq 0}$ on a homogeneous Poisson point process $χ$ on $\R^d$ ($d\geq 1$), starts at the origin and at each step picks its next Poisson point among its closest neighbors according to i.i.d. labels having the same distribution as $K$. Our main result (Theorem 1) states that the number of Poisson points visited by $(X_n)_{n \geq 0}$ admits an exponential decay whenever the random variable $K$ has a bounded support (BS). In particular, the $K$-th nearest neighbor random walk visits finitely many Poisson points if and only if $K$ satisfies Assumption (BS). To prove it, we introduce the key notion of pioneer point which allows us to deal with the region of $\R^d$ already explored by $(X_n)_{n \geq 0}$. Still under Assumption (BS), we also prove an exponential decay for the Euclidean length of the trajectory performed by $(X_n)_{n \geq 0}$ (Theorem 2). Finally, and quite surprisingly, we exhibit an example of label distribution with bounded support for which the $K$-th nearest neighbor random walk discovers new Poisson points after a number of steps whose tail distribution is at least polynomial (Theorem 3).

math.PR↗
arXiv2025

First-order behavior of the time constant in non-isotropic continuous first-passage percolation

Consider $Ξ$ a homogeneous Poisson point process on $\mathbb{R}^d$ ($d\geq 2$) with unit intensity with respect to the Lebesgue measure. For $\varepsilon\geq 0$, we define the Boolean model $Σ_{p, \varepsilon}$ as the union of the balls of volume $\varepsilon$ for the $p$-norm ($p\in [1,\infty]$) and centered at the points of $Ξ$. We define a random pseudo-metric on $\mathbb{R}^d$ by associating with any path a travel time equal to its $p$-length outside $Σ_{p,\varepsilon}$. This defines a continuous model of first-passage percolation, that has been studied in \cite{GT17,GT22} for $p=2$, the Euclidean norm. For $p=1$, this model is expected to share common properties with the classical first-passage percolation on the graph $\mathbb{Z}^d$ with a distribution of passage times of the form $\varepsilon δ_0 + (1-\varepsilon) δ_1$. The exact calculation of the time constant of this model $\tilde μ_{p,\varepsilon} (x)$ is out of reach. We investigate here the behavior of $\varepsilon \mapsto \tilde μ_{p,\varepsilon} (x)$ near $0$, and enlighten how the speed at which $\| x \|_p - \tilde μ_{p,\varepsilon} (x) $ goes to $0$ depends on $x$ and $p$. For instance, for $p\in (1,\infty)$, we prove that $\| x \|_p - \tilde μ_{p,ε} (x)$ is of order $\varepsilon ^{κ_p(x)}$ with $$κ_p(x): = \frac{1}{d- \frac{d_1(x)-1}{2} - \frac{d-d_1 (x)}{p}}\,,$$where $d_1(x)$ is the number of non null coordinates of $x$. The exact order of $\| x \|_p - \tilde μ_{p,ε} (x)$ is also given for $p=1$ and $p=\infty$. Related results are also discussed, about properties of the geodesics, and analog properties on closely related models.

math.PR↗
arXiv2021

First-order behavior of the time constant in Bernoulli first-passage percolation

We consider the standard model of first-passage percolation on $\mathbb{Z}^d$ ($d\geq 2$), with i.i.d. passage times associated with either the edges or the vertices of the graph. We focus on the particular case where the distribution of the passage times is the Bernoulli distribution with parameter $1-ε$. These passage times induce a random pseudo-metric $T_ε$ on $\mathbb{R}^d$. By subadditive arguments, it is well known that for any $z\in\mathbb{R}^d\setminus \{0\}$, the sequence $T_ε(0,\lfloor nz \rfloor) / n$ converges a.s. towards a constant $μ_ε(z)$ called the time constant. We investigate the behavior of $ε\mapsto μ_ε(z)$ near $0$, and prove that $μ_ε(z) = \| z\|_1 - C (z) ε^{1/d_1(z)} + o ( ε^{1/d_1(z)}) $, where $d_1(z)$ is the number of non null coordinates of $z$, and $C(z)$ is a constant whose dependence on $z$ is partially explicit.

math.PR↗
arXiv2021

Large deviation principle for the streams and the maximal flow in first passage percolation

We consider the standard first passage percolation model in the rescaled lattice $\mathbb{Z}^d$ for $d\geq 2$ and a bounded domain $Ω$ in $\mathbb R ^d$. We denote by $Γ^1$ and $Γ^2$ two disjoint subsets of $\partial Ω$ representing respectively the source and the sink, i.e., where the water can enter in $Ω$ and escape from $Ω$. A maximal stream is a vector measure $\overrightarrowμ_n^{max}$ that describes how the maximal amount of fluid can enter through $Γ^1$ and spreads in $Ω$. Under some assumptions on $Ω$ and $G$, we already know a law of large number for $\overrightarrowμ_n^{max}$. The sequence $(\overrightarrowμ_n^{max})_{n\geq 1} $ converges almost surely to the set of solutions of a continuous deterministic problem of maximal stream in an anisotropic network. We aim here to derive a large deviation principle for streams and deduce by contraction principle the existence of a rate function for the upper large deviations of the maximal flow in $Ω$.

math.PR↗
arXiv2021

Large deviation principle for the cutsets and lower large deviation principle for the maximal flow in first passage percolation

We consider the standard first passage percolation model in the rescaled lattice $\mathbb Z^d/n$ for $d\geq 2$ and a bounded domain $Ω$ in $\mathbb R^d$. We denote by $Γ^1$ and $Γ^2$ two disjoint subsets of $\partial Ω$ representing respectively the sources and the sinks, \textit{i.e.}, where the water can enter in $Ω$ and escape from $Ω$. A cutset is a set of edges that separates $Γ^1$ from $Γ^2$ in $Ω$, it has a capacity given by the sum of the capacities of its edges. Under some assumptions on $Ω$ and the distribution of the capacities of the edges, we already know a law of large numbers for the sequence of minimal cutsets $(\mathcal E_n^{min})_{n\geq 1}$: the sequence $(\mathcal E_n^{min})_{n\geq 1}$ converges almost surely to the set of solutions of a continuous deterministic problem of minimal cutset in an anisotropic network. We aim here to derive a large deviation principle for cutsets and deduce by contraction principle a lower large deviation principle for the maximal flow in $Ω$.

math.PR↗
arXiv2020

Continuity of the time constant in a continuous model of first passage percolation

For a given dimension d $\ge$ 2 and a finite measure $ν$ on (0, +$\infty$), we consider $ξ$ a Poisson point process on R d x (0, +$\infty$) with intensity measure dc $\otimes$ $ν$ where dc denotes the Lebesgue measure on R d. We consider the Boolean model $Σ$ = $\cup$ (c,r)$\in$$ξ$ B(c, r) where B(c, r) denotes the open ball centered at c with radius r. For every x, y $\in$ R d we define T (x, y) as the minimum time needed to travel from x to y by a traveler that walks at speed 1 outside $Σ$ and at infinite speed inside $Σ$. By a standard application of Kingman sub-additive theorem, one easily shows that T (0, x) behaves like $μ$ x when x goes to infinity, where $μ$ is a constant named the time constant in classical first passage percolation. In this paper we investigate the regularity of $μ$ as a function of the measure $ν$ associated with the underlying Boolean model.

math.PR↗
arXiv2018

Existence and continuity of the flow constant in first passage percolation

We consider the model of i.i.d. first passage percolation on Z^d, where we associate with the edges of the graph a family of i.i.d. random variables with common distribution G on [0, +$\infty$] (including +$\infty$). Whereas the time constant is associated to the study of 1-dimensional paths with minimal weight, namely geodesics, the flow constant is associated to the study of (d--1)-dimensional surfaces with minimal weight. In this article, we investigate the existence of the flow constant under the only hypothesis that G({+$\infty$}) < p c (d) (in particular without any moment assumption), the convergence of some natural maximal flows towards this constant, and the continuity of this constant with regard to the distribution G.

math.PR↗
arXiv2018

Size of a minimal cutset in supercritical first passage percolation

We consider the standard model of i.i.d. first passage percolation on Z^d given a distribution G on [0, +$\infty$] (including +$\infty$). We suppose that G({0}) > 1 -- p\_c(d), i.e., the edges of positive passage time are in the subcritical regime of percolation on Z^d. We consider a cylinder of basis an hyperrectangle of dimension d -- 1 whose sides have length n and of height h(n) with h(n) negligible compared to n (i.e., h(n)/n $\rightarrow$ 0 when n goes to infinity). We study the maximal flow from the top to the bottom of this cylinder. We already know that the maximal flow renormalized by n^(d--1) converges towards the flow constant which is null in the case G({0}) > 1 -- p\_c (d). The study of maximal flow is associated with the study of sets of edges of minimal capacity that cut the top from the bottom of the cylinder. If we denote by $ψ$\_n the minimal cardinal of such a set of edges, we prove here that $ψ$\_n /n^(d--1) converges almost surely towards a constant.

math.PR↗
arXiv2018

Equivalence of some subcritical properties in continuum percolation

We consider the Boolean model on $\R^d$. We prove some equivalences between subcritical percolation properties. Let us introduce some notations to state one of these equivalences. Let $C$ denote the connected component of the origin in the Boolean model. Let $|C|$ denotes its volume. Let $\ell$ denote the maximal length of a chain of random balls from the origin. Under optimal integrability conditions on the radii, we prove that $E(|C|)$ is finite if and only if there exists $A,B >0$ such that $¶(\ell \ge n) \le Ae^{-Bn}$ for all $n \ge 1$.

math.PR↗
arXiv2017

Positivity of the time constant in a continuous model of first passage percolation

We consider a non trivial Boolean model $Σ$ on ${\mathbb R}^d$ for $d\geq 2$. For every $x,y \in {\mathbb R}^d$ we define $T(x,y)$ as the minimum time needed to travel from $x$ to $y$ by a traveler that walks at speed $1$ outside $Σ$ and at infinite speed inside $Σ$. By a standard application of Kingman sub-additive theorem, one easily shows that $T(0,x)$ behaves like $μ\|x\|$ when $\|x\|$ goes to infinity, where $μ$ is a constant named the time constant in classical first passage percolation. In this paper we investigate the positivity of $μ$. More precisely, under an almost optimal moment assumption on the radii of the balls of the Boolean model, we prove that $μ\textgreater{}0$ if and only if the intensity $λ$ of the Boolean model satisfies $λ\textless{} \widehatλ\_c$, where $ \widehatλ\_c$ is one of the classical critical parameters defined in continuum percolation.

math.PR↗
arXiv2016

Continuity of the time and isoperimetric constants in supercritical percolation

We consider two different objects on super-critical Bernoulli percolation on $\mathbb{Z}^d$ : the time constant for i.i.d. first-passage percolation (for $d\geq 2$) and the isoperimetric constant (for $d=2$). We prove that both objects are continuous with respect to the law of the environment. More precisely we prove that the isoperimetric constant of supercritical percolation in $\mathbb{Z}^2$ is continuous in the percolation parameter. As a corollary we prove that normalized sets achieving the isoperimetric constant are continuous with respect to the Hausdroff metric. Concerning first-passage percolation, equivalently we consider the model of i.i.d. first-passage percolation on $\mathbb{Z}^d$ with possibly infinite passage times: we associate with each edge $e$ of the graph a passage time $t(e)$ taking values in $[0,+\infty]$, such that $\mathbf{P}[t(e)<+\infty] >p_c(d)$. We prove the continuity of the time constant with respect to the law of the passage times. This extends the continuity property previously proved by Cox and Kesten for first passage percolation with finite passage times.

math.PR↗
arXiv2014

Weak shape theorem in first passage percolation with infinite passage times

We consider the model of i.i.d. first passage percolation on $\mathbb{Z}^d$ : we associate with each edge $e$ of the graph a passage time $t(e)$ taking values in $[0,+\infty]$, such that $\mathbb{P}[t(e)<+\infty] >p_c(d)$. Equivalently, we consider a standard (finite) i.i.d. first passage percolation model on a super-critical Bernoulli percolation performed independently. We prove a weak shape theorem without any moment assumption. We also prove that the corresponding time constant is positive if and only if $\mathbb{P}[t(e)=0]<p_c(d)$.

math.PR↗
arXiv2014

Maximal stream and minimal cutset for first passage percolation through a domain of $\mathbb{R}^d$

We consider the standard first passage percolation model in the rescaled graph $\mathbb{Z}^d/n$ for $d\geq2$ and a domain $Ω$ of boundary $Γ$ in $\mathbb{R}^d$. Let $Γ^1$ and $Γ^2$ be two disjoint open subsets of $Γ$, representing the parts of $Γ$ through which some water can enter and escape from $Ω$. A law of large numbers for the maximal flow from $Γ^1$ to $Γ^2$ in $Ω$ is already known. In this paper we investigate the asymptotic behavior of a maximal stream and a minimal cutset. A maximal stream is a vector measure $\vecμ_n^{\max}$ that describes how the maximal amount of fluid can cross $Ω$. Under conditions on the regularity of the domain and on the law of the capacities of the edges, we prove that the sequence $(\vecμ_n^{\max})_{n\geq1}$ converges a.s. to the set of the solutions of a continuous deterministic problem of maximal stream in an anisotropic network. A minimal cutset can been seen as the boundary of a set $E_n^{\min}$ that separates $Γ^1$ from $Γ^2$ in $Ω$ and whose random capacity is minimal. Under the same conditions, we prove that the sequence $(E_n^{\min})_{n\geq1}$ converges toward the set of the solutions of a continuous deterministic problem of minimal cutset. We deduce from this a continuous deterministic max-flow min-cut theorem and a new proof of the law of large numbers for the maximal flow. This proof is more natural than the existing one, since it relies on the study of maximal streams and minimal cutsets, which are the pertinent objects to look at.

math.PR↗
arXiv2012

Upper large deviations for the maximal flow through a domain of $\bolds{\mathbb{R}^d}$ in first passage percolation

We consider the standard first passage percolation model in the rescaled graph $\mathbb {Z}^d/n$ for $d\geq2$ and a domain $Ω$ of boundary $Γ$ in $\mathbb {R}^d$. Let $Γ^1$ and $Γ^2$ be two disjoint open subsets of $Γ$ representing the parts of $Γ$ through which some water can enter and escape from $Ω$. We investigate the asymptotic behavior of the flow $ϕ_n$ through a discrete version $Ω_n$ of $Ω$ between the corresponding discrete sets $Γ^1_n$ and $Γ^2_n$. We prove that under some conditions on the regularity of the domain and on the law of the capacity of the edges, the upper large deviations of $ϕ_n/n^{d-1}$ above a certain constant are of volume order, that is, decays exponentially fast with $n^d$. This article is part of a larger project in which the authors prove that this constant is the a.s. limit of $ϕ_n/n^{d-1}$.

math.PR↗
arXiv2010

Law of large numbers for the maximal flow through tilted cylinders in two-dimensional first passage percolation

Equip the edges of the lattice $\mathbb{Z}^2$ with i.i.d. random capacities. We prove a law of large numbers for the maximal flow crossing a rectangle in $\mathbb{R}^2$ when the side lengths of the rectangle go to infinity. The value of the limit depends on the asymptotic behaviour of the ratio of the height of the cylinder over the length of its basis. This law of large numbers extends the law of large numbers obtained by Grimmett and Kesten (1984) for rectangles of particular orientation.

math.PR↗
arXiv2009

Lower large deviations for the maximal flow through tilted cylinders in two-dimensional first passage percolation

Equip the edges of the lattice $\mathbb{Z}^2$ with i.i.d. random capacities. A law of large numbers is known for the maximal flow crossing a rectangle in $\mathbb{R}^2$ when the side lengths of the rectangle go to infinity. We prove that the lower large deviations are of surface order, and we prove the corresponding large deviation principle from below. This extends and improves previous large deviations results of Grimmett and Kesten (1984) obtained for boxes of particular orientation.

math.PR↗
arXiv2009

Lower large deviations for the maximal flow through a domain of $\mathbb{R}^d$ in first passage percolation

We consider the standard first passage percolation model in the rescaled graph $\mathbb{Z}^d/n$ for $d\geq 2$, and a domain $Ω$ of boundary $Γ$ in $\mathbb{R}^d$. Let $Γ^1$ and $Γ^2$ be two disjoint open subsets of $Γ$, representing the parts of $Γ$ through which some water can enter and escape from $Ω$. We investigate the asymptotic behaviour of the flow $ϕ_n$ through a discrete version $Ω_n$ of $Ω$ between the corresponding discrete sets $Γ^1_n$ and $Γ^2_n$. We prove that under some conditions on the regularity of the domain and on the law of the capacity of the edges, the lower large deviations of $ϕ_n/ n^{d-1}$ below a certain constant are of surface order.

math.PR↗