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Algebraic generating functions for two-dimensional random walks

Abstract : In this paper, we caracterize the solutions of specific bi-variate functional equations. The unknown functions represent the steady state distribution of specific two-dimensional random walks on Z2+. Inside the quarter plane, the jump have amplitude one, but are arbitrary on the axes. The main result is a necessary and sufficient condition for the solutions to be algebraic, when the "group" of the random walk (associated to an algebraic curve Q(x,y) = 0 having genus one) is finite. The method is based on Hilbert's factorization theorems, together with a uniformisation by means of elliptic functions.
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Submitted on : Wednesday, May 24, 2006 - 6:05:22 PM
Last modification on : Friday, February 4, 2022 - 3:22:05 AM
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  • HAL Id : inria-00075374, version 1



Guy Fayolle, Roudolph Iasnogorodski, Vadim A. Malyshev. Algebraic generating functions for two-dimensional random walks. RR-1184, INRIA. 1990. ⟨inria-00075374⟩



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