Classifying lattice walks restricted to the quarter plane

This work considers the nature of generating functions of random lattice walks restricted to the first quadrant. In particular, we find combinatorial criteria to decide if related series are algebraic, transcendental holonomic or otherwise. Complete results for walks taking their steps in a maximum of three directions of restricted amplitude are given, as is a well-supported conjecture for all walks with steps taken from a subset of 2{0,±1}. New enumerative results are presented for several classes, each obtained with a variant of the kernel method.