Asymptotic Betti Numbers for Hard Squares in the Homological Liquid Regime

Abstract

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 $j$th Betti number for large $n$, $j$, $p$, and $q$. We consider sequences of area-normalized coordinates, where $\left (\frac {n}{pq}, \frac {j}{pq}\right )$ 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 identified in [3], 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.