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- W2968275738 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 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." @default.
- W2968275738 created "2019-08-22" @default.
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- W2968275738 date "2019-08-12" @default.
- W2968275738 modified "2023-09-27" @default.
- W2968275738 title "Configuration spaces of disks in an infinite strip" @default.
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