Area-width scaling in generalised Motzkin paths

TL;DR

This paper studies a generalized model of Motzkin paths with variable horizontal step length, deriving functional equations and analyzing the scaling behavior of the area-length generating function, including rigorous analysis near critical points.

Contribution

It introduces a generalized Motzkin path model with fixed horizontal step length and derives its functional equations and scaling behavior, extending previous special cases.

Findings

Heuristic ansatz accurately predicts scaling for Dyck and Schr"oder paths.

Derived explicit generating functions for Schr"oder paths.

Rigorous analysis near tri-critical points confirms heuristic predictions.

Abstract

We consider a generalised version of Motzkin paths, where horizontal steps have length $\ell$, with $\ell$ being a fixed positive integer. We first give the general functional equation for the area-length generating function of this model. Using a heuristic ansatz, we derive the area-length scaling behaviour in terms of a scaling function in one variable for the special cases of Dyck, (standard) Motzkin and Schr\"oder paths, before generalising our approach to arbitrary $\ell$. We then derive an expression for the generating function of Schr\"oder paths and analyse the scaling behaviour of this function rigorously in the vicinity of the tri-critical point of the model by applying the method of steepest descents for the case of two coalescing saddle points. Our results show that for Dyck and Schr\"oder paths, the heuristic scaling ansatz reproduces the rigorous results.