On the functions counting walks with small steps in the quarter plane

By Irina A. Kurkova, Kilian Raschel. Publications mathématiques (Bures-sur-Yvette), 2011

84 citations recorded by the source corpus.

Models of spatially homogeneous walks in the quarter plane \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbf{ Z}_{+}^{2}$\end{document} with steps taken from a subset \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{S}$\end{document} of the set of jumps to the eight nearest neighbors are considered. The generating function (x,y,z)↦Q(x,y;z) of the numbers q(i,j;n) of such walks starting at the origin and ending at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(i,j) \in\mathbf{ Z}_{+}^{2}$\end{document} after n steps is studied. For all non-singular models of walks, the functions x↦Q(x,0;z) and y↦Q(0,y;z) are continued as multi-valued functions on C having infinitely many meromorphic branches, of which the set of poles is identified. The nature of these functions is derived from this result: namely, for all the 51 walks which admit a certain infinite group of birational transformations of C2, the interval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$]0,1/|\mathcal{S}|[$\end{document} of variation of z splits into two dense subsets such that the functions x↦Q(x,0;z) and y↦Q(0,y;z) are shown to be holonomic for any z from the one of them and non-holonomic for any z from the other. This entails the non-holonomy of (x,y,z)↦Q(x,y;z), and therefore proves a conjecture of Bousquet-Mélou and Mishna in Contemp. Math. 520:1–40 (2010).

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ef9bf0150fc57d9127120400372e98996db764c5

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Citations in the corpus

On the nature of the generating series of walks in the quarter plane (2017, methodology): “ by W and is described in Figure 1. This Figure reproduces the table [KR12, Figure 17] and uses notations compatible with loc. cit. for the convenience of the reader and for the ease of reference. In [KR12], the authors show that the generating series for all walks in W are nonholonomic. In [BBMR16]”
Random walks in cones (2011, result): “MR1691900 [20] Fayolle, G. and Raschel, K. (2012).”