arXiv subjects

Julien Poisat

Publications and source records attributed to Julien Poisat.

At least 19 recordsLinked to original sources

arXiv2026

Strong large deviation principles for pair empirical measures of random walks in the Mukherjee-Varadhan topology

In this paper we introduce a topology under which the pair empirical measure of a large class of random walks satisfies a strong Large Deviation principle. The definition of the topology is inspired by the recent article by Mukherjee and Varadhan~\cite{MV2016}. This topology is natural for translation-invariant problems such as the downward deviations of the volume of a Wiener sausage or simple random walk, known as the Swiss cheese model~\cite{BBH2001}. We also adapt our result to some rescaled random walks and provide a contraction principle to the single empirical measure despite a lack of continuity from the projection map, using the notion of diagonal tightness.

math.PR↗

arXiv2025

A connection between the random pinning model and random walks in sparse random environments

The purpose of this short note is to establish a connection between a one-dimensional random walk in a random sparse environment and the random pinning model. We show that the grand canonical partition function of the pinning model coincides with the mean number of returns to the origin for a random walk in a random sparse environment averaged on the randomness location. We obtain thereof some information on the integrability of the number of return times in the annealed and partially annealed setups.

math.PR↗

arXiv2025

Variational representation and estimates for the free energy of a quenched charged polymer model

Random walks with a disordered self-interaction potential may be used to model charged polymers. In this paper we consider a one-dimensional and directed version of the charged polymer model that was introduced by Derrida, Griffiths and Higgs. We prove new results for the associated quenched free energy, including a variational formula based on a quenched large deviation principle established by Birkner, Greven and den Hollander. We also take the occasion to (i) provide detailed proofs for state-of-the-art results pointing towards the existence of a freezing transition and (ii) proceed with minor corrections for two results previously obtained by the present author with Caravenna, den Hollander and P{é}tr{é}lis for the undirected model.

math.PR↗

arXiv2025

Phase diagram of the interacting partially directed self-avoiding walk attracted by a vertical wall

In the present paper, we consider the interacting partially-directed self-avoiding walk (IPDSAW) attracted by a vertical wall. The IPDSAW was introduced by Zwanzig and Lauritzen (J. Chem. Phys., 1968) as a manner of investigating the collapse transition of a homopolymer dipped in a repulsive solvent. We prove in particular that a surface transition occurs inside the collapsed phase between (i) a regime where the attractive vertical wall does not influence the geometry of the polymer and (ii) a regime where the polymer is partially attached at the wall on a length that is comparable to its horizontal extension, modifying its asymptotic Wulff shape. The latter rigorously confirms the conjecture exposed by physicists in (Physica A: Stat. Mech. \\& App., 2002). We push the analysis even further by providing sharp asymptotics of the partition function inside the collapsed phase.

math.PR↗

arXiv2024

Localization of a one-dimensional simple random walk among power-law renewal obstacles

We consider a one-dimensional simple random walk killed by quenched soft obstacles. The position of the obstacles is drawn according to a renewal process with a power-law increment distribution. In a previous work, we computed the large-time asymptotics of the quenched survival probability. In the present work we continue our study by describing the behaviour of the random walk conditioned to survive. We prove that with large probability, the walk quickly reaches a unique time-dependent optimal gap that is free from obstacle and gets localized there. We actually establish a dichotomy. If the renewal tail exponent is smaller than one then the walk hits the optimal gap and spends all of its remaining time inside, up to finitely many visits to the bottom of the gap. If the renewal tail exponent is larger than one then the random walk spends most of its time inside of the optimal gap but also performs short outward excursions, for which we provide matching upper and lower bounds on their length and cardinality. Our key tools include a Markov renewal interpretation of the survival probability as well as various comparison arguments for obstacle environments. Our results may also be rephrased in terms of localization properties for a directed polymer among multiple repulsive interfaces.

math.PR↗

arXiv2023

Uniqueness and tube property for the Swiss cheese large deviations

We consider the simple random walk on the Euclidean lattice, in three dimensions and higher, conditioned to visit fewer sites than expected, when the deviation from the mean scales like the mean. The associated large deviation principle was first derived in 2001 by van den Berg, Bolthausen and den Hollander in the continuous setting, that is for the volume of a Wiener sausage, and later taken up by Phetpradap in the discrete setting. One of the key ideas in their work is to condition the range of the random walk to a certain skeleton, that is a sub-sequence of the random walk path taken along an appropriate mesoscopic scale. In this paper we prove that (i) the rate function obtained by van den Berg, Bolthausen and den Hollander has a unique minimizer over the set of probability measures modulo shifts, at least for deviations of the range well below the mean, and (ii) the empirical measure of the skeleton converges under the conditioned law, in a certain manner, to this minimizer. To this end we use an adaptation of the topology recently introduced by Mukherjee and Varadhan to compactify the space of probability measures.

math.PR↗

arXiv2018

The random pinning model with correlated disorder given by a renewal set

We investigate the effect of correlated disorder on the localization transition undergone by a renewal sequence with loop exponent $α$ > 0, when the correlated sequence is given by another independent renewal set with loop exponent $α$ > 0. Using the renewal structure of the disorder sequence, we compute the annealed critical point and exponent. Then, using a smoothing inequality for the quenched free energy and second moment estimates for the quenched partition function, combined with decoupling inequalities, we prove that in the case $α$ > 2 (summable correlations), disorder is irrelevant if $α$ < 1/2 and relevant if $α$ > 1/2, which extends the Harris criterion for independent disorder. The case $α$ $\in$ (1, 2) (non-summable correlations) remains largely open, but we are able to prove that disorder is relevant for $α$ > 1/ $α$, a condition that is expected to be non-optimal. Predictions on the criterion for disorder relevance in this case are discussed. Finally, the case $α$ $\in$ (0, 1) is somewhat special but treated for completeness: in this case, disorder has no effect on the quenched free energy, but the annealed model exhibits a phase transition.

math.PR↗

arXiv2018

A limit theorem for the survival probability of a simple random walk among power-law renewal traps

We consider a one-dimensional simple random walk surviving among a field of static soft traps : each time it meets a trap the walk is killed with probability 1--e --$β$ , where $β$ is a positive and fixed parameter. The positions of the traps are sampled independently from the walk and according to a renewal process. The increments between consecutive traps, or gaps, are assumed to have a power-law decaying tail with exponent $γ$ > 0. We prove convergence in law for the properly rescaled logarithm of the quenched survival probability as time goes to infinity. The normalization exponent is $γ$/($γ$ + 2), while the limiting law writes as a variational formula with both universal and non-universal features. The latter involves (i) a Poisson point process that emerges as the universal scaling limit of the properly rescaled gaps and (ii) a function of the parameter $β$ that we call asymptotic cost of crossing per trap and that may, in principle, depend on the details of the gap distribution. Our proof suggests a confinement strategy of the walk in a single large gap. This model may also be seen as a (1 + 1)-directed polymer among many repulsive interfaces, in which case $β$ corresponds to the strength of repulsion, the survival probability to the partition function and its logarithm to the finite-volume free energy. Along the way we prove a stochastic monotonicity property for the hitting time of the killed random walk with respect to the non-killed one, that could be of interest in other contexts, see Proposition 3.5.

math.PR↗

arXiv2017

Annealed scaling for a charged polymer in dimensions two and higher

This paper considers an undirected polymer chain on $\mathbb{Z}^d$, $d \geq 2$, with i.i.d.\ random charges attached to its constituent monomers. Each self-intersection of the polymer chain contributes an energy to the interaction Hamiltonian that is equal to the product of the charges of the two monomers that meet. The joint probability distribution for the polymer chain and the charges is given by the Gibbs distribution associated with the interaction Hamiltonian. The object of interest is the \emph{annealed free energy} per monomer in the limit as the length $n$ of the polymer chain tends to infinity. We show that there is a critical curve in the parameter plane spanned by the charge bias and the inverse temperature separating an \emph{extended phase} from a \emph{collapsed phase}. We derive the scaling of the critical curve for small and for large charge bias and the scaling of the annealed free energy for small inverse temperature, which are both anomalous. We show that in a subset of the collapsed phase the polymer chain is \emph{subdiffusive}, namely, on scale $(n/\log n)^{1/(d+2)}$ it moves like a Brownian motion conditioned to stay inside a ball with a deterministic radius and a randomly shifted center. We expect this scaling to hold throughout the collapsed phase. We further expect that in the extended phase the polymer chain scales like a weakly self-avoiding walk. Proofs are based on a detailed analysis for simple random walk of the downward large deviations of the self-intersection local time and the upward large deviations of the range. Part of our scaling results are rough, and we formulate conjectures under which they can be sharpened. The existence of the free energy remains an open problem, which we are able to settle in a subset of the collapsed phase for a subclass of charge distributions.

math-ph↗

arXiv2016

Asymptotics of the critical time in Wiener sausage percolation with a small radius

We consider a continuum percolation model on $\R^d$, where $d\geq 4$.The occupied set is given by the union of independent Wiener sausages with radius $r$ running up to time $t$ and whoseinitial points are distributed according to a homogeneous Poisson point process.It was established in a previous work by Erhard, Martínez and Poisat~\cite{EMP13} that (1) if $r$ is small enough there is a non-trivial percolation transitionin $t$ occuring at a critical time $t\_c(r)$ and (2) in the supercritical regime the unbounded cluster is unique. In this paper we investigate the asymptotic behaviour of the critical time when the radius $r$ converges to $0$. The latter does not seem to be deducible from simple scaling arguments. We prove that for $d\geq 4$, there is a positive constant $c$ such that$c^{-1}\sqrt{\log(1/r)}\leq t\_c(r)\leq c\sqrt{\log(1/r)}$ when $d=4$ and $c^{-1}r^{(4-d)/2}\leq t\_c(r) \leq c\ r^{(4-d)/2}$ when $d\geq 5$, as $r$ converges to $0$. We derive along the way moment estimates on the capacity of Wiener sausages, which may be of independent interest.

math.PR↗

arXiv2015

Brownian Paths Homogeneously Distributed in Space: Percolation Phase Transition and Uniqueness of the Unbounded Cluster

We consider a continuum percolation model on $\R^d$, $d\geq 1$.For $t,λ\in (0,\infty)$ and $d\in\{1,2,3\}$, the occupied set is given by the union of independent Brownian paths running up to time $t$ whoseinitial points form a Poisson point process with intensity $λ\textgreater{}0$.When $d\geq 4$, the Brownian paths are replaced by Wiener sausageswith radius $r\textgreater{}0$.We establish that, for $d=1$ and all choices of $t$, no percolation occurs,whereas for $d\geq 2$, there is a non-trivial percolation transitionin $t$, provided $λ$ and $r$ are chosen properly.The last statement means that $λ$ has to be chosen to be strictly smaller than the critical percolation parameter for the occupied set at time zero(which is infinite when $d\in\{2,3\}$, but finite and dependent on $r$ when $d\geq 4$).We further show that for all $d\geq 2$, the unbounded cluster in the supercritical phase is unique.Along the way a finite box criterion for non-percolation in the Boolean model is extended to radius distributions with an exponential tail. This may be of independent interest.The present paper settles the basic properties of the model and should be viewed as a jumpboard for finer results.

math.PR↗

arXiv2014

On the critical curves of the Pinning and Copolymer models in Correlated Gaussian environment

We investigate the disordered copolymer and pinning models, in the case of a correlated Gaussian environment with summable correlations, and when the return distribution of the underlying renewal process has a polynomial tail. As far as the copolymer model is concerned, we prove disorder relevance both in terms of critical points and critical exponents, in the case of non-negative correlations. When some of the correlations are negative, even the annealed model becomes non-trivial. Moreover, when the return distribution has a finite mean, we are able to compute the weak coupling limit of the critical curves for both models, with no restriction on the correlations other than summability. This generalizes the result of Berger, Caravenna, Poisat, Sun and Zygouras \cite{cf:BCPSZ} to the correlated case. Interestingly, in the copolymer model, the weak coupling limit of the critical curve turns out to be the maximum of two quantities: one generalizing the limit found in the IID case \cite{cf:BCPSZ}, the other one generalizing the so-called Monthus bound.

math.PR↗

arXiv2013

Large deviation principles for words drawn from correlated letter sequences

When an i.i.d.\ sequence of letters is cut into words according to i.i.d.\ renewal times, an i.i.d.\ sequence of words is obtained. In the \emph{annealed} LDP (large deviation principle) for the empirical process of words, the rate function is the specific relative entropy of the observed law of words w.r.t.\ the reference law of words. In Birkner, Greven and den Hollander \cite{BGdH10} the \emph{quenched} LDP (= conditional on a typical letter sequence) was derived for the case where the renewal times have an \emph{algebraic} tail. The rate function turned out to be a sum of two terms, one being the annealed rate function, the other being proportional to the specific relative entropy of the observed law of letters w.r.t.\ the reference law of letters, obtained by concatenating the words and randomising the location of the origin. The proportionality constant equals the tail exponent of the renewal process. The purpose of the present paper is to extend both LDP's to letter sequences that are not i.i.d. It is shown that both LDP's carry over when the letter sequence satisfies a mixing condition called \emph{summable variation}. The rate functions are again given by specific relative entropies w.r.t.\ the reference law of words, respectively, letters. But since neither of these reference laws is i.i.d., several approximation arguments are needed to obtain the extension.

math.PR↗

arXiv2013

A Quenched Functional Central Limit Theorem for Planar Random Walks in Random Sceneries

Random walks in random sceneries (RWRS) are simple examples of stochastic processes in disordered media. They were introduced at the end of the 70's by Kesten-Spitzer and Borodin, motivated by the construction of new self-similar processes with stationary increments. Two sources of randomness enter in their definition: a random field $ξ= (ξ_x)_{x \in \Z^d}$ of i.i.d.\ random variables, which is called the \emph{random scenery}, and a random walk $S = (S_n)_{n \in \N}$ evolving in $\Z^d$, independent from the scenery. The RWRS $Z = (Z_n)_{n \in \N}$ is then defined as the accumulated scenery along the trajectory of the random walk, i.e., $Z_n := \sum_{k=1}^n ξ_{S_k}$. The law of $Z$ under the joint law of $ξ$ and $S$ is called "annealed", and the conditional law given $ξ$ is called "quenched". Recently, central limit theorems under the quenched law were proved for $Z$ by the first two authors for a class of transient random walks including walks with finite variance in dimension $d \ge 3$. In this paper we extend their results to dimension $d=2$.

math.PR↗

arXiv2013

The Critical Curve of the Random Pinning and Copolymer Models at Weak Coupling

We study random pinning and copolymer models, when the return distribution of the underlying renewal process has a polynomial tail with finite mean. We compute the asymptotic behavior of the critical curves of the models in the weak coupling regime, showing that it is universal. This proves a conjecture of Bolthausen, den Hollander and Opoku for copolymer models (ref. [8]), which we also extend to pinning models.

math.PR↗

arXiv2012

Ruelle-Perron-Frobenius operator approach to the annealed pinning model with Gaussian long-range correlated disorder

In this paper we study the pinning model with correlated Gaussian disorder. The presence of correlations makes the annealed model more involved than the usual homogeneous model, which is fully solvable. We prove however that if the disorder correlations decay fast enough then the annealed critical behaviour is the same as the homogeneous one. Our result is sharper if the decay is exponential. The approach we propose relies on the spectral properties of a transfer or Ruelle-Perron Frobenius operator related to the model. We use results on these operators that were obtained in the framework of the thermodynamic formalism for countable Markov shifts. We also provide large-temperature asymptotics of the annealed critical curve under weaker assumptions.

math.PR↗

arXiv2012

Quenched Central Limit Theorems for Random Walks in Random Scenery

Random walks in random scenery are processes defined by $$Z_n:=\sum_{k=1}^nω_{S_k}$$ where $S:=(S_k,k\ge 0)$ is a random walk evolving in $\mathbb{Z}^d$ and $ω:=(ω_x, x\in{\mathbb Z}^d)$ is a sequence of i.i.d. real random variables. Under suitable assumptions on the random walk $S$ and the random scenery $ω$, almost surely with respect to $ω$, the correctly renormalized sequence $(Z_n)_{n\geq 1}$ is proved to converge in distribution to a centered Gaussian law with explicit variance.

math.PR↗

arXiv2012

Random pinning model with finite range correlations : disorder relevant regime

The purpose of this paper is to show how one can extend some results on disorder relevance obtained for the random pinning model with i.i.d disorder to the model with finite range correlated disorder. In a previous work, the annealed critical curve of the latter model was computed, and equality of quenched and annealed critical points, as well as exponents, was proved under some conditions on the return exponent of the interarrival times. Here we complete this work by looking at the disorder relevant regime, where annealed and quenched critical points differ. All these results show that the Harris criterion, which was proved to be correct in the i.i.d case, remains valid in our setup. We strongly use Markov renewal constructions that were introduced in the solving of the annealed model.

math.PR↗