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Reversed Dirichlet environment and directional transience of random walks in Dirichlet environment

Christophe SabotLaurent Tournier — 2011

Annales de l'I.H.P. Probabilités et statistiques

We consider random walks in a random environment given by i.i.d. Dirichlet distributions at each vertex of ℤ or, equivalently, oriented edge reinforced random walks on ℤ. The parameters of the distribution are a 2-uplet of positive real numbers indexed by the unit vectors of ℤ. We prove that, as soon as these weights are nonsymmetric, the random walk is transient in a direction (i.e., it satisfies  ⋅  → +∞ for some ) with positive probability. In dimension 2, this result is strenghened...

Stokes matrices of hypergeometric integrals

Alexey GlutsyukChristophe Sabot — 2010

Annales de l’institut Fourier

In this work we compute the Stokes matrices of the ordinary differential equation satisfied by the hypergeometric integrals associated to an arrangement of hyperplanes in generic position. This generalizes the computation done by J.-P. Ramis for confluent hypergeometric functions, which correspond to the arrangement of two points on the line. The proof is based on an explicit description of a base of canonical solutions as integrals on the cones of the arrangement, and combinatorial relations between...

Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model

Christophe SabotPierre Tarrès — 2015

Journal of the European Mathematical Society

Edge-reinforced random walk (ERRW), introduced by Coppersmith and Diaconis in 1986 [8], is a random process which takes values in the vertex set of a graph G and is more likely to cross edges it has visited before. We show that it can be represented in terms of a vertex-reinforced jump process (VRJP) with independent gamma conductances; the VRJP was conceived by Werner and first studied by Davis and Volkov [10, 11], and is a continuous-time process favouring sites with more local time. We calculate,...

Aging and quenched localization for one-dimensional random walks in random environment in the sub-ballistic regime

Nathanaël EnriquezChristophe SabotOlivier Zindy — 2009

Bulletin de la Société Mathématique de France

We consider transient one-dimensional random walks in a random environment with zero asymptotic speed. An aging phenomenon involving the generalized Arcsine law is proved using the localization of the walk at the foot of “valleys“ of height log t . In the quenched setting, we also sharply estimate the distribution of the walk at time t .

Limit laws for transient random walks in random environment on

Nathanaël EnriquezChristophe SabotOlivier Zindy — 2009

Annales de l’institut Fourier

We consider transient random walks in random environment on with zero asymptotic speed. A classical result of Kesten, Kozlov and Spitzer says that the hitting time of the level n converges in law, after a proper normalization, towards a positive stable law, but they do not obtain a description of its parameter. A different proof of this result is presented, that leads to a complete characterization of this stable law. The case of Dirichlet environment turns out to be remarkably explicit.

Diffusion and propagation problems in some ramified domains with a fractal boundary

Yves AchdouChristophe SabotNicoletta Tchou — 2006

ESAIM: Mathematical Modelling and Numerical Analysis

This paper is devoted to some elliptic boundary value problems in a self-similar ramified domain of 2 with a fractal boundary. Both the Laplace and Helmholtz equations are studied. A generalized Neumann boundary condition is imposed on the fractal boundary. Sobolev spaces on this domain are studied. In particular, extension and trace results are obtained. These results enable the investigation of the variational formulation of the above mentioned boundary value problems. Next, for homogeneous...

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