Staircase polygons: moments of diagonal lengths and column heights

TL;DR

This paper analyzes staircase polygons by their perimeter and geometric properties, deriving limit distributions for diagonal lengths and column heights, and compares theoretical results with Monte Carlo simulations.

Contribution

It introduces new methods to derive limit distributions for staircase polygons' geometric parameters and applies these to related models of directed walks.

Findings

Limit distributions for diagonal lengths and column heights are established.

Theoretical results align well with Monte Carlo simulations.

Methods are applicable to related models of directed walks.

Abstract

We consider staircase polygons, counted by perimeter and sums of k-th powers of their diagonal lengths, k being a positive integer. We derive limit distributions for these parameters in the limit of large perimeter and compare the results to Monte-Carlo simulations of self-avoiding polygons. We also analyse staircase polygons, counted by width and sums of powers of their column heights, and we apply our methods to related models of directed walks.