Non-Generic Analytic Combinatorics in Several Variables and Lattice Path Asymptotics

Abstract

Finding and analyzing generating functions is a classic and powerful way of studying combinatorial structures. While generating functions are a priori purely formal objects, there is a rich theory treating them as complex-analytic functions. This is the field of analytic combinatorics, which exploits classical results in complex and harmonic analysis to find asymptotics of coefficient sequences of generating functions. More recently, the field of analytic combinatorics in several variables has been developed to study multivariate generating functions, which are amenable to multivariate analytic results.

When dealing with multivariate asymptotics we first pick a direction vector. To find asymptotics using analytic combinatorics, one typically uses the Cauchy integral formula to write the underlying sequence being studied as a complex integral, then uses residue computations to reduce to a so-called Fourier-Laplace integral. When the direction vector chosen is generic, the Fourier-Laplace integral being studied is well-behaved and asymptotics can be computed. When the direction vector chosen is non-generic, the Fourier-Laplace integral has singular amplitude, and asymptotics are harder to compute. This thesis studies singular Fourier-Laplace integrals and their applications to combinatorics.

Determining asymptotics for the number of lattice walks restricted to certain regions is one of the particular successes of analytic combinatorics in several variables. In particular, Melczer and Mishna determined asymptotics for the number of walks in an orthant whose set of steps is a subset of { ±1, 0 }^d \ { 0 } and is symmetric over every axis. Melczer and Wilson later determined asymptotics for lattice path models whose set of steps is a subset of { ±1, 0 }^d \ { 0 } and is symmetric over all but one axis, except when the vector sum of all the steps is 0. Here we determine asymptotics in the zero sum case using asymptotics of singular Fourier-Laplace integrals. We additionally study asymptotics of lattice path models restricted to Weyl chambers, before giving some generalizations of asymptotics of singular Fourier-Laplace integrals.