Overall Objectives
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Highlights of the Year
New Software and Platforms
New Results
Bilateral Contracts and Grants with Industry
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Section: New Results

Random Walks in Orthants

Participant : Guy Fayolle.

Explicit criterion for the finiteness of the group in the quarter plane

In the book [3] , original methods were proposed to determine the invariant measure of random walks in the quarter plane with small jumps, the general solution being obtained via reduction to boundary value problems. Among other things, an important quantity, the so-called group of the walk, allows to deduce theoretical features about the nature of the solutions. In particular, when the order of the group is finite, necessary and sufficient conditions have been given in [3] for the solution to be rational or algebraic. When the underlying algebraic curve is of genus 1, we propose, in collaboration with R. Iasnogorodski (St-Petersburg, Russia), a concrete criterion ensuring the finiteness of the group. It turns out that this criterion is always tantamount to the cancellation of a single constant, which can be expressed as the determinant of a matrix of order 3 or 4, and depends in a polynomial way on the coefficients of the walk [20] .

Second Edition of the Book Random walks in the Quarter Plane

In collaboration with R. Iasnogorodski (St-Petersburg, Russia) and V. Malyshev, we prepared the second edition of the book [3] , which will be published by Springer, in the collection Probability Theory and Stochastic Processes. Part II of this second edition borrows specific case-studies from queueing theory, and enumerative combinatorics. Five chapters will be added, including examples and applications of the general theory to enumerative combinatorics. Among them: