Variants of the Kernel Method for Lattice Path Models

The kernel method has proved to be an extremely versatile tool for exact and asymptotic enumeration. Recent applications in the study of lattice walks have linked combinatorial properties of a model to algebraic conditions on its generating function, demonstrating how to extract additional information from the process. This thesis details two new results. In the first, we apply the iterated kernel method to determine asymptotic information about a family of models in the quarter plane, finding their generating functions explicitly and classifying them as non D-finite. The second considers d-dimensional walks restricted to an octant whose step sets are symmetric over every axis. A generalized version of the orbit sum method allows for a representation of their generating functions as diagonals of multivariate rational functions, proving they are D-finite. In combination with current developments from analytic combinatorics in several variables, this yields dominant asymptotics for all such models.