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Nadine Guillotin-Plantard

Publications and source records attributed to Nadine Guillotin-Plantard.

At least 19 recordsLinked to original sources
arXiv2025

Iterated random walks in random scenery (PAPAPA)

We establish a limit theorem for a new model of 3-dimensional random walk in an inhomogeneous lattice with random orientations. This model can be seen as a 3dimensional version of the Matheron and de Marsily model [12]. This new model leads us naturally to the study of iterated random walk in random scenery, which is a new process that can be described as a random walk in random scenery evolving in a second random scenery. We use the french acronym PAPAPA for this new process and answer a question about its stochastic behaviour asked about twenty years ago by St{é}phane Le Borgne.

math.PR↗
arXiv2019

Empirical processes for recurrent and transient random walks in random scenery

In this paper, we are interested in the asymptotic behaviour of the sequence of processes $(W_n(s,t))_{s,t\in[0,1]}$ with \begin{equation*} W_n(s,t):=\sum_{k=1}^{\lfloor nt\rfloor}\big(1_{\{ξ_{S_k}\leq s\}}-s\big) \end{equation*} where $(ξ_x, x\in\mathbb{Z}^d)$ is a sequence of independent random variables uniformly distributed on $[0,1]$ and $(S_n)_{n\in\mathbb N}$ is a random walk evolving in $\mathbb{Z}^d$, independent of the $ξ$'s. In Wendler (2016), the case where $(S_n)_{n\in\mathbb N}$ is a recurrent random walk in $\mathbb{Z}$ such that $(n^{-\frac 1α}S_n)_{n\geq 1}$ converges in distribution to a stable distribution of index $α$, with $α\in(1,2]$, has been investigated. Here, we consider the cases where $(S_n)_{n\in\mathbb N}$ is either: a) a transient random walk in $\mathbb{Z}^d$, b) a recurrent random walk in $\mathbb{Z}^d$ such that $(n^{-\frac 1d}S_n)_{n\geq 1}$ converges in distribution to a stable distribution of index $d\in\{1,2\}$.

math.PR↗
arXiv2019

Persistence probabilities for stationary increment processes

We study the persistence probability for processes with stationary increments. Our results apply to a number of examples: sums of stationary correlated random variables whose scaling limit is fractional Brownian motion, random walks in random sceneries, random processes in Brownian scenery, and the Matheron-de Marsily model in Z^2 with random orientations of the horizontal layers. Using a new approach, strongly related to the study of the range, we obtain an upper bound of optimal order in the general case and improved lower bounds (compared to previous literature) for many processes.

math.PR↗
arXiv2019

Branching processes in correlated random environment

We consider the critical branching processes in correlated random environment which is positively associated and study the probability of survival up to the n-th generation. Moreover, when the environment is given by fractional Brownian motion, we estimate also the tail of progeny as well as the tail of width.

math.PR↗
arXiv2016

Random walks and branching processes in correlated Gaussian environment

We study persistence probabilities for random walks in correlated Gaussian random environment first studied by Oshanin, Rosso and Schehr. From the persistence results, we can deduce properties of critical branching processes with offspring sizes geometrically distributed with correlated random parameters. More precisely, we obtain estimates on the tail distribution of its total population size, of its maximum population, and of its extinction time.

math.PR↗
arXiv2015

On the survival probability in the Matheron - De Marsily model

We are interested in the behaviour of the range and of the first return time to the origin of random walks in random scenery. As a byproduct a precise estimate of the survival probability in the Matheron and de Marsily model is obtained. Our result confirms the conjectures announced by Majumdar and Redner.

math.PR↗
arXiv2015

The quenched limiting distributions of a charged-polymer model

The limit distributions of the charged-polymer Hamiltonian of Kantor and Kardar [Bernoulli case] and Derrida, Griffiths and Higgs [Gaussian case] are considered. Two sources of randomness enter in the definition: a random field $q= (q_i)_{i\geq 1}$ of i.i.d. random variables (called random charges) and a random walk $S = (S_n)_{n \in \mathbb{N}}$ evolving in $\mathbb{Z}^d$, independent of the charges. The energy or Hamiltonian $K = (K_n)_{n \geq 2}$ is then defined as $$K_n := \sum_{1\leq i < j\leq n} q_i q_j {\bf 1}_{\{S_i=S_j\}}.$$ The law of $K$ under the joint law of $q$ and $S$ is called "annealed", and the conditional law given $q$ is called "quenched". Recently, strong approximations under the annealed law were proved for $K$. In this paper we consider the limit distributions of $K$ under the quenched law.

math.PR↗
arXiv2015

Persistence exponent for random walk on directed versions of $Z^2$

We study the persistence exponent for random walks in random sceneries (RWRS) with integer values and for some special random walks in random environment in $\mathbb Z^2$ including random walks in $\mathbb Z^2$ with random orientations of the horizontal layers.

math.PR↗
arXiv2015

On the range of the Campanino and Pétritis random walk

We are interested in the behaviour of the range of the Campanino and Pétritis random walk, namely a simple random walk on the lattice $\mathbb Z^2$ with random orientations of the horizontal layers. We also study the range of random walks in random scenery, from which the asymptotic behaviour of the range of the first coordinate of the Campanino and Pétritis random walk can be deduced.

math.PR↗
arXiv2015

Persistence exponent for random processes in Brownian scenery

In this paper we consider the persistence properties of random processes in Brownian scenery, which are examples of non-Markovian and non-Gaussian processes. More precisely we study the asymptotic behaviour for large $T$, of the probability $P[ \sup\_{t\in[0,T]} Δ\_t \leq 1] $ where $Δ\_t = \int\_{\mathbb{R}} L\_t(x) \, dW(x).$ Here $W={W(x); x\in\mathbb{R}}$ is a two-sided standard real Brownian motion and ${L\_t(x); x\in\mathbb{R},t\geq 0}$ is the local time of some self-similar random process $Y$, independent from the process $W$. We thus generalize the results of \cite{BFFN} where the increments of $Y$ were assumed to be independent.

math.PR↗
arXiv2015

Persistence exponent for discrete-time, time-reversible processes

We study the persistence probability for some discrete-time, time-reversible processes. In particular, we deduce the persistence exponent in a number of examples: first, we deal with random walks in random sceneries (RWRS) in any dimension with Gaussian scenery. Second, we deal with sums of stationary Gaussian sequences with correlations exhibiting long-range dependence. Apart from the persistence probability we deal with the position of the maximum and the time spent on the positive half-axis by the process.

math.PR↗
arXiv2014

A combinatorial approach to a model of constrained random walkers

In [1], the authors consider a random walk $(Z_{n,1},\ldots,Z_{n,K+1})\in \mathbb{Z}^{K+1}$ with the constraint that each coordinate of the walk is at distance one from the following one. A functional central limit theorem for the first coordinate is proved and the limit variance is explicited. In this paper, we study an extended version of this model by conditioning the extremal coordinates to be at some fixed distance at every time. We prove a functional central limit theorem for this random walk. Using combinatorial tools, we give a precise formula of the variance and compare it with the one obtained in [1].

math.PR↗
arXiv2013

Central Limit Theorems for Open Quantum Random Walks and Quantum Measurement Records

Open Quantum Random Walks, as developed in \cite{APSS}, are a quantum generalization of Markov chains on finite graphs or on lattices. These random walks are typically quantum in their behavior, step by step, but they seem to show up a rather classical asymptotic behavior, as opposed to the quantum random walks usually considered in Quantum Information Theory (such as the well-known Hadamard random walk). Typically, in the case of Open Quantum Random Walks on lattices, their distribution seems to always converge to a Gaussian distribution or a mixture of Gaussian distributions. In the case of nearest neighbors homogeneous Open Quantum Random Walks on $\ZZ^d$ we prove such a Central Limit Theorem, in the case where only one Gaussian distribution appears in the limit. Through the quantum trajectory point of view on quantum master equations, we transform the problem into studying a certain functional of a Markov chain on $\ZZ^d$ times the Banach space of quantum states. The main difficulty is that we know nothing about the invariant measures of this Markov chain, even their existence. Surprisingly enough, we are able to produce a Central Limit Theorem with explicit drift and explicit covariance matrix. The interesting point which appears with our construction and result is that it applies actually to a wider setup: it provides a Central Limit Theorem for the sequence of recordings of the quantum trajectories associated to any completely positive map. This is what we show and develop as an application of our result. In a second step we are able to extend our Central Limit Theorem to the case of several asymptotic Gaussians, in the case where the operator coefficients of the quantum walk are block-diagonal in a common basis.

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↗
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↗
arXiv2011

Renewal theorems for random walks in random scenery

Random walks in random scenery are processes defined by $Z_n:=\sum_{k=1}^nξ_{X_1+...+X_k}$, where $(X_k,k\ge 1)$ and $(ξ_y,y\in\mathbb Z)$ are two independent sequences of i.i.d. random variables. We suppose that the distributions of $X_1$ and $ξ_0$ belong to the normal domain of attraction of strictly stable distributions with index $α\in[1,2]$ and $β\in(0,2)$ respectively. We are interested in the asymptotic behaviour as $|a|$ goes to infinity of quantities of the form $\sum_{n\ge 1}{\mathbb E}[h(Z_n-a)]$ (when $(Z_n)_n$ is transient) or $\sum_{n\ge 1}{\mathbb E}[h(Z_n)-h(Z_n-a)]$ (when $(Z_n)_n$ is recurrent) where $h$ is some complex-valued function defined on $\mathbb{R}$ or $\mathbb{Z}$.

math.PR↗
arXiv2010

On the local time of random walks associated with Gegenbauer polynomials

The local time of random walks associated with Gegenbauer polynomials $P_n^{(α)}(x),\ x\in [-1,1]$ is studied in the recurrent case: $α\in\ [-\frac{1}{2},0]$. When $α$ is nonzero, the limit distribution is given in terms of a Mittag-Leffler distribution. The proof is based on a local limit theorem for the random walk associated with Gegenbauer polynomials. As a by-product, we derive the limit distribution of the local time of some particular birth and death Markov chains on $\bbN$.

math.PR↗