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Olivier Garet

Publications and source records attributed to Olivier Garet.

At least 19 recordsLinked to original sources
arXiv2026

Probabilistic proof for non-survival at criticality : the Galton-Watson process and more

In a famous paper, Bezuidenhout and Grimmett demonstrated that the contact process dies out at the critical point.Their proof technique has often been used to study the growth of population patterns. The present text is intended as an introduction to their ideas, with examples of minimal technicality. In particular, we recover the basic theorem about Galton-Watson chains: except in a degenerate case, survival is possible only if the fertility rate exceeds 1. The classical proof that is taught in classrooms is essentially analytic, based on generating functions and convexity arguments. Following the Bezuidenhout-Grimmett way, we propose a proof that is more consistent with probabilistic intuition. We also study the survival problem for a cooperative model, mixing sexual and asexual reproduction.

math.PR↗
arXiv2025

A simple master Theorem for discrete divide and conquer recurrences

The aim of this note is to provide a Master Theorem for some discrete divide and conquer recurrences: $$X_{n}=a_n+\sum_{j=1}^m b_j X_{\lfloor{\frac{n}{m_j}}\rfloor},$$ where the $m_i$'s are integers with $m_i\ge 2$. The main novelty of this work is there is no assumption of regularity or monotonicity for $(a_n)$. Then, this result can be applied to various sequences of random variables $(a_n)_{n\ge 0}$, for example such that $\sup_{n\ge 1}\mathbb{E}(|a_n|)<+\infty$.

math.CA↗
arXiv2025

How often is $x\mapsto x^3$ one-to-one in $\mathbb{Z}/n\mathbb{Z}$?

We characterize the integers n such that $x\mapsto x^3$ describes a bijection from the set $\mathbb{Z}/n\mathbb{Z}$ to itself and we determine the frequency of these integers. Precisely, denoting by $W$ the set of these integers, we prove that an integer belongs to $W$ if and only if it is square-free with no prime factor that is congruent to 1 modulo 3, and that there exists $C>0$ such that $$|W\cap\{1,\dots,n\}|\sim C\frac{n}{\sqrt{\log n}}\ .$$ These facts (or equivalent facts) are stated without proof on the OEIS website. We give the explicit value of $C$, which did not seem to be known. Analogous results are also proved for families of integers for which congruence classes for prime factors are imposed. The proofs are based on a Tauberian Theorem by Delange.

math.NT↗
arXiv2023

Arithm{é}tique des primes pluri-annuelles

This short note studies the simple arithmetic properties of multi-year bonus allocation systems, as recently implemented in the population of teacher-researchers, with the PEDR on the one hand, and with component 3 of the RIPEC on the other.In the first stage, we study an initial model in which the entire population applies, in order to highlight the effect of the multi-year nature of the award: the success rate of an application is very different from the frequency with which an individual receives the bonus.Secondly, we take into account the fact that part of the population is not a candidate, with cross-influences between the competition participation rate and the individual success rate.In the light of available statistics on agent behavior, we will discuss the possibility of RIPEC training a large part of the population.

physics.soc-ph↗
arXiv2021

Does Eulerian percolation on $Z^2$ percolate ?

Eulerian percolation on Z 2 with parameter p is the classical Bernoulli bond percolation with parameter p conditioned on the fact that every site has an even degree. We first explain why Eulerian percolation with parameter p coincides with the contours of the Ising model for a well-chosen parameter $β$(p). Then we study the percolation properties of Eulerian percolation.

math.PR↗
arXiv2021

Percolation and first-passage percolation on oriented graphs

We give the first properties of independent Bernoulli percolation, for oriented graphs on the set of vertices $\Z^d$ that are translation-invariant and may contain loops. We exhibit some examples showing that the critical probability for the existence of an infinite cluster may be direction-dependent. Then, we prove that the phase transition in a given direction is sharp, and study the links between percolation and first-passage percolation on these oriented graphs.

math.PR↗
arXiv2016

The critical Galton-Watson process vanishes

The Galton-Watson process belongs to standard teaching in probability. The basic theorem says that survival is only possible when the fecondity exceeds 1. The classical proof is essentially analytic, using generating functions and convexity arguments. We give here a more probabilistic proof, in the spirit of Bezuidenhout and Grimmett.

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↗
arXiv2016

Les lois Zêta pour l'arithmétique

This paper provides a probabilist point of view about some results in analytic number theory. The main tool is the family of Zeta laws, which is a consolation for the non-existence of an uniform law on the set of integers. We prove the existence and compute the natural density for the pairs of coprime integers, and also for the pairs of coprime Gaussian integers.Along the way, we recover the decomposition of the Zeta function as an Eulerian product and some related results.

math.PR↗
arXiv2015

The Number of Open Paths in Oriented Percolation

We study the number $N\_n$ of open paths of length $n$ in supercritical oriented percolation on $\Zd \times \N$, with $d \ge 1$. We prove that on the percolation event $\{\inf N\_n\textgreater{}0\}$, $N\_n^{1/n}$ almost surely converges to a positive deterministic constant. We also study the existence of directional limits. The proof relies on the introduction of adapted sequences of regenerating times, on subadditive arguments and on the properties of the coupled zone in supercritical oriented percolation.

math.PR↗
arXiv2013

Growth of a population of bacteria in a dynamical hostile environment

We study the growth of a population of bacteria in a dynamical hostile environment corresponding to the immune system of the colonised organism. The immune cells evolve as subcritical open clusters of oriented percolation and are perpetually reinforced by an immigration process, while the bacteria try to grow as a supercritical oriented percolation in the remaining empty space. For appropriate values of the parameters, we prove that the population of bacteria grows linearly. In this perspective, we build general tools to study dependent percolation models issued from renormalization processes.

q-bio.PE↗
arXiv2012

The critical branching random walk in a random environment dies out

We study the possibility for branching random walks in random environment (BRWRE) to survive. The particles perform simple symmetric random walks on the $d$-dimensional integer lattice, while at each time unit, they split into independent copies according to time-space i.i.d. offspring distributions. As noted by Comets and Yoshida, the BRWRE is naturally associated with the directed polymers in random environment (DPRE), for which the quantity $Ψ$ called the free energy is well studied. Comets and Yoshida proved that there is no survival when $Ψ<0$ and that survival is possible when $Ψ>0$. We proved here that, except for degenerate cases, the BRWRE always die when $Ψ=0$. This solves a conjecture of Comets and Yoshida.

math.PR↗
arXiv2012

Asymptotic shape for the contact process in random environment

The aim of this article is to prove asymptotic shape theorems for the contact process in stationary random environment. These theorems generalize known results for the classical contact process. In particular, if H_t denotes the set of already occupied sites at time t, we show that for almost every environment, when the contact process survives, the set H_t/t almost surely converges to a compact set that only depends on the law of the environment. To this aim, we prove a new almost subadditive ergodic theorem.

math.PR↗
arXiv2012

Bacterial persistence: a winning strategy?

It has long been known that antibiotic treatment will not completely kill off a bacteria population. For many species a small fraction of bacteria is not sensitive to antibiotics. These bacteria are said to persist. Recently it has been shown that persistence is not a permanent state and that in fact a bacterium can switch back and forth between persistent and non persistent states. We introduce two stochastic models for bacteria persistence. In both models there are mass killings of non persistent bacteria at certain times. The first model has deterministic killing times and the second one has random killing times. Both models suggest that persistence may be a successful strategy for a wide range of parameter values.

math.PR↗
arXiv2012

Large deviations for the contact process in random environment

The asymptotic shape theorem for the contact process in random environment gives the existence of a norm $μ$ on $\Rd$ such that the hitting time $t(x)$ is asymptotically equivalent to $μ(x)$ when the contact process survives. We provide here exponential upper bounds for the probability of the event $\{\frac{t(x)}{μ(x)}\not\in [1-ε,1+ε]\}$; these bounds are optimal for independent random environment. As a special case, this gives the large deviation inequality for the contact process in a deterministic environment, which, as far as we know, has not been established yet.

math.PR↗
arXiv2009

La forme asymptotique du processus de contact en environnement aléatoire

The aim of this article is to prove asymptotic shape theorems for the contact process in stationary random environment. These theorems generalize known results for the classical contact process. In particular, if H_t denotes the set of already occupied sites at time t, we show that for almost every environment, when the contact process survives, the set H_t/t almost surely converges to a compact set that only depends on the law of the environment. To this aim, we prove a new almost subadditive ergodic theorem.

math.PR↗