Asymptotic Betti numbers for hard squares in the homological liquid regime

TL;DR

This paper investigates the asymptotic behavior of Betti numbers in configuration spaces of ordered squares within rectangles, revealing they grow factorially with the number of squares and identifying a homological liquid regime.

Contribution

It establishes the factorial growth rate of Betti numbers for large configurations and characterizes the homological liquid regime in the parameter space.

Findings

Betti numbers grow factorially with the number of squares

The homological liquid regime covers all points in the feasible region

Betti numbers are significantly larger than those for point configurations

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 $(\frac{n}{pq}, \frac{j}{pq})$ 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.