Walks in the quarter plane: Analytic approach and applications

Abstract

Abstract. In this survey we present an analytic approach to solve problems concerning (deterministic or random) walks in the quarter plane. We illustrate the recent breakthroughs in that domain with two examples. The first one is about the combinatorics of walks confined to the quarter plane, and more precisely about the numbers of walks evolving in the quarter plane and having given length, starting and ending points. We show how to obtain exact and asymptotic expressions for these numbers, and how to find the algebraic nature of their generating function. The second example deals with population biology, and more specifically with the extinction probabilities of certain flower populations.-6