Limit laws for discrete excursions and meanders and linear functional equations with a catalytic variable

TL;DR

This paper investigates the limit distributions of certain combinatorial and probabilistic models using algebraic and analytic techniques, revealing connections to Brownian motion and deriving explicit joint distributions.

Contribution

It introduces a novel approach combining the method of moments with the kernel method to analyze limit laws for discrete excursions, meanders, and related structures.

Findings

Limit distributions include Brownian excursion and meander area distributions.

Explicit joint distributions of area and final altitude for various walk models.

Computed area laws for discrete excursions, meanders, and polyominoes.

Abstract

We study limit distributions for random variables defined in terms of coefficients of a power series which is determined by a certain linear functional equation. Our technique combines the method of moments with the kernel method of algebraic combinatorics. As limiting distributions the area distributions of the Brownian excursion and meander occur. As combinatorial applications we compute the area laws for discrete excursions and meanders with an arbitrary finite set of steps and the area distribution of column convex polyominoes. As a by-product of our approach we find the joint distribution of area and final altitude for meanders with an arbitrary step set, and for unconstrained Bernoulli walks (and hence for Brownian Motion) the joint distribution of signed areas and final altitude. We give these distributions in terms of their moments.