Asymptotic Betti numbers for hard squares in the homological liquid regime

2022/07/26 by Hannah Alpert, Alpert, Hannah, Matthew Kahle +3 · 1 citation
Chemistry · Computer Science · Mathematics · #55R80 (82B26) #Algebraic Topology (math.AT) #Chromatography in Natural Products #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2207.13139

openalex publication_date 2022/07/26 · openalex created_date 2022/07/30 · openalex updated_date 2026/07/28

Abstract

We study configuration spaces C(n; p, q) of n ordered unit squares in a p by q rectangle. Our goal is to estimate the Betti numbers for large n, j, p, and q. We consider sequences of area-normalized coordinates, where ((n)/(pq), (j)/(pq)) converges as n, j, p, and q approach infinity. For every sequence that converges to a point in the "feasible region" in the (x,y)-plane, we show that the factorial growth rate of the Betti numbers is the same as the factorial growth rate of n!. This implies that (1) the Betti numbers are vastly larger than for the configuration space of n ordered points in the plane, which have the factorial growth rate of j!, and (2) every point in the feasible region is eventually in the homological liquid regime.

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