arXiv subjects

Guy Fayolle

Publications and source records attributed to Guy Fayolle.

24 records · Page 2Linked to original sources

arXiv2010

On the Holonomy or Algebraicity of Generating Functions Counting Lattice Walks in the Quarter-Plane

In two recent works \cite{BMM,BK}, it has been shown that the counting generating functions (CGF) for the 23 walks with small steps confined in a quadrant and associated with a finite group of birational transformations are holonomic, and even algebraic in 4 cases -- in particular for the so-called Gessel's walk. It turns out that the type of functional equations satisfied by these CGF appeared in a probabilistic context almost 40 years ago. Then a method of resolution was proposed in \cite{FIM}, involving at once algebraic tools and a reduction to boundary value problems. Recently this method has been developed in a combinatorics framework in \cite{Ra}, where a thorough study of the explicit expressions for the CGF is proposed. The aim of this paper is to derive the nature of the bivariate CGF by a direct use of some general theorems given in \cite{FIM}.

math.PR↗

arXiv2008

On the Almost Sure Central Limit Theorem for Vector Martingales: Convergence of Moments and Statistical Applications

We investigate the almost sure asymptotic properties of vector martingale transforms. Assuming some appropriate regularity conditions both on the increasing process and on the moments of the martingale, we prove that normalized moments of any even order converge in the almost sure cental limit theorem for martingales. A conjecture about almost sure upper bounds under wider hypotheses is formulated. The theoretical results are supported by examples borrowed from statistical applications, including linear autoregressive models and branching processes with immigration, for which new asymptotic properties are established on estimation and prediction errors.

math.PR↗

arXiv2006

Stochastic Dynamics of Discrete Curves and Multi-type Exclusion Processes

This study deals with continuous limits of interacting one-dimensional diffusive systems, arising from stochastic distortions of discrete curves with various kinds of coding representations. These systems are essentially of a reaction-diffusion nature. In the non-reversible case, the invariant measure has generally a non Gibbs form. The corresponding steady-state regime is analyzed in detail with the help of a tagged particle and a state-graph cycle expansion of the probability currents. As a consequence, the constants appearing in Lotka-Volterraequations --which describe the fluid limits of stationary states-- can be traced back directly at the discrete level to tagged particles cycles coefficients. Current fluctuations are also studied and the Lagrangian is obtained by an iterative scheme. The related Hamilton-Jacobi equation, which leads to the large deviation functional, is analyzed and solved in the reversible case for the sake of checking.

cond-mat.stat-mech↗

arXiv2006

Stochastic Deformations of Sample Paths of Random Walks and Exclusion Models

This study in centered on models accounting for stochastic deformations of sample paths of random walks, embedded either in $\mathbb{Z}^2$ or in $\mathbb{Z}^3$. These models are immersed in multi-type particle systems with exclusion. Starting from examples, we give necessary and sufficient conditions for the underlying Markov processes to be reversible, in which case their invariant measure has a Gibbs form. Letting the size of the sample path increase, we find the convenient scalings bringing to light phase transition phenomena. Stable and metastable configurations are bound to time-periods of limiting deterministic trajectories which are solution of nonlinear differential systems: in the example of the ABC model, a system of Lotka-Volterra class is obtained, and the periods involve elliptic, hyper-elliptic or more general functions. Lastly, we discuss briefly the contour of a general approach allowing to tackle the transient regime via differential equations of Burgers' type.

cond-mat.stat-mech↗

arXiv2006

Stochastic Dynamics of Discrete Curves and Exclusion Processes. Part 1: Hydrodynamic Limit of the ASEP System

This report is the foreword of a series dedicated to stochastic deformations of curves. Problems are set in terms of exclusion processes, the ultimate goal being to derive hydrodynamic limits for these systems after proper scalings. In this study, solely the basic \textsc{asep} system on the torus is analyzed. The usual sequence of empirical measures, converges in probability to a deterministic measure, which is the unique weak solution of a Cauchy problem. The method presents some new features, letting hope for extensions to higher dimension. It relies on the analysis of a family of parabolic differential operators, involving variational calculus. Namely, the variables are the values of functions at given points, their number being possibly infinite.

math.PR↗

arXiv2002

Dynamical Windings of Random Walks and Exclusion Models. Part I: Thermodynamic Limit

We consider a system consisting of a planar random walk on a square lattice, submitted to stochastic elementary local deformations. Depending on the deformation transition rates, and specifically on a parameter $η$ which breaks the symmetry between the left and right orientation, the winding distribution of the walk is modified, and the system can be in three different phases: folded, stretched and glassy. An explicit mapping is found, leading to consider the system as a coupling of two exclusion processes. For all closed or periodic initial sample paths, a convenient scaling permits to show a convergence in law (or almost surely on a modified probability space) to a continuous curve, the equation of which is given by a system of two non linear stochastic differential equations. The deterministic part of this system is explicitly analyzed via elliptic functions. In a similar way, by using a formal fluid limit approach, the dynamics of the system is shown to be equivalent to a system of two coupled Burgers' equations.

cond-mat.stat-mech↗