Analytic Combinatorics of Planar Lattice Paths

TL;DR

This paper surveys and develops combinatorial methods for analyzing the asymptotic enumeration of planar lattice paths, especially in the quarter plane, with applications in physics, chemistry, and probability.

Contribution

It introduces a novel systematic combinatorial approach for directly determining exponential growth factors of lattice paths in restricted regions.

Findings

Developed a new method for asymptotic enumeration of lattice paths

Provided explicit formulas for growth rates in quarter plane paths

Connected lattice path enumeration with entropy in physical systems

Abstract

Lattice paths effectively model phenomena in chemistry, physics and probability theory. Asymptotic enumeration of lattice paths is linked with entropy in the physical systems being modeled. Lattice paths restricted to different regions of the plane are well suited to a functional equation approach for exact and asymptotic enumeration. This thesis surveys results on lattice paths under various restrictions, with an emphasis on lattice paths in the quarter plane. For these paths, we develop an original systematic combinatorial approach providing direct access to the exponential growth factors of the asymptotic expressions.