On the evolution of random integer compositions

TL;DR

This paper investigates the asymptotic structural evolution of random integer compositions as the total sum increases, identifying thresholds for various subpatterns and highlighting a dichotomy in pattern appearance and disappearance.

Contribution

It provides a detailed analysis of the thresholds for the emergence and vanishing of specific substructures in random integer compositions, revealing new insights into their probabilistic behavior.

Findings

Thresholds identified for the appearance of subpatterns

Disappearance points for various runs and patterns

Dichotomy between the emergence of small and large patterns

Abstract

We explore how the asymptotic structure of a random $n$-term weak integer composition of $m$ evolves, as $m$ increases from zero. The primary focus is on establishing thresholds for the appearance and disappearance of substructures. These include the longest and shortest runs of zero terms or of nonzero terms, longest increasing runs, longest runs of equal terms, largest squares (runs of $k$ terms each equal to $k$), as well as a wide variety of other patterns. Of particular note is the dichotomy between the appearance and disappearance of exact consecutive patterns, with smaller patterns appearing before larger ones, whereas longer patterns disappear before shorter ones.