2023/09/12 by David Bevan, Bevan, David, Dan Threlfall +1
Mathematics · #05A05 #05C80 #60C05 #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics
paper · doi:10.48550/arxiv.2309.06287
openalex publication_date 2023/09/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.