# Configuration spaces of disks in an infinite strip

## TL;DR

This paper analyzes the topology of configuration spaces of hard disks in an infinite strip, identifying different regimes where homology behaves distinctly, revealing polynomial growth, exponential growth, and phase transitions in the homological structure.

## Contribution

It characterizes the homology of disk configuration spaces in an infinite strip across different width regimes, linking topological behavior to phase transition analogies.

## Key findings

- Homology $H_j[C(n,w)]$ matches point configuration space when $w \\ge j+2$.
- Betti numbers grow polynomially or exponentially depending on the width regime.
- Identifies homological phase transitions analogous to solid, liquid, and gas states.

## Abstract

We study the topology of the configuration spaces $C(n,w)$ of $n$ hard disks of unit diameter in an infinite strip of width $w$. We describe ranges of parameter or "regimes", where homology $H_j [C(n,w)]$ behaves in qualitatively different ways. We show that if $w \ge j+2$, then the homology $H_j[C(n, w)]$ is isomorphic to the homology of the configuration space of points in the plane, $H_j[C(n, \mathbb{R}^2)]$. The Betti numbers of $C(n, \mathbb{R}^2) $ were computed by Arnold, and so as a corollary of the isomorphism, $\beta_j[C(n,w)]$ is a polynomial in $n$ of degree $2j$. On the other hand, we show that if $2 \le w \le j+1$, then $\beta_j [ C(n,w) ]$ grows exponentially with $n$. Most of our work is in carefully estimating $\beta_j [ C(n,w) ]$ in this regime. We also illustrate, for every $n$, the homological "phase portrait" in the $(w,j)$-plane--- the parameter values where homology $H_j [C(n,w)]$ is trivial, nontrivial, and isomorphic with $H_j [C(n, \mathbb{R}^2)]$. Motivated by the notion of phase transitions for hard-spheres systems, we discuss these as the "homological solid, liquid, and gas" regimes.