2025/09/17 by Filippo Colomo, Colomo, Filippo, A. G. Pronko +1
Mathematics · #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Random Matrices and Applications #Statistical Mechanics (cond-mat.stat-mech)
paper · pdf · doi:10.48550/arxiv.2509.14006
openalex publication_date 2025/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
An Alternating Sign Matrix (ASM) is a square matrix with entries in \0,1,-1\, and such that: i) in each row and columns, nonzero entries alternate in sign; ii) for any given row or column, entries sum up to 1. We define the frozen-square enumeration as the enumeration of n× n ASMs under the refinement of having, located in a corner, an s× s square of entries that are all zeroes. We state a conjectural formula for such enumeration, in terms of the determinant of some s× s matrix whose entries are given explicitly. We provide numerical support in favour of our conjecture. We also illustrate the relevance of the conjectured formula in connection with the limit shape observed in large ASMs, its fluctuations, and the Tracy--Widom distribution.