Interacting Quarter-Plane Lattice Walk Problems: Solutions and Proofs

Abstract

Lattice walk problems in the quarter-plane have been widely studied in recent years. The main objective is to calculate the number of configurations, that is the number of $n$-step walks ending at certain points or alternatively, the generating function of the walks. In combinatorics, physics and probability theory, other properties such as asymptotic behavior and the algebra of the generating functions are also of interest. In this thesis we focus on solving quarter-plane lattice walks with interactions via the kernel method. We assign interaction $a$ to the $x$-axis, $b$ to the $y$-axis and $c$ to the origin. We denote $q_{n,k,l,h,u,v..