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arXiv · 1709.03111

On percolation of two-dimensional hard disks

Alexander Magazinov
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Abstract

We consider the hard-core model in $\mathbb{R}^2$, in which a random set of non-intersecting unit disks is sampled with an intensity parameter $λ$. Given $\varepsilon>0$ we consider the graph in which two disks are adjacent if they are at distance $\leq \varepsilon$ from each other. We prove that this graph, $G$, is highly connected when $λ$ is greater than a certain threshold depending on $\varepsilon$. Namely, given a square annulus with inner radius $L_1$ and outer radius $L_2$, the probability that the annulus is crossed by $G$ is at least $1 - C \exp(-cL_1)$. As a corollary we prove that a Gibbs state admits an infinite component of $G$ if the intensity $λ$ is large enough, depending on $\varepsilon$.

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Alexander Magazinov. 2017-09-10. On percolation of two-dimensional hard disks. https://doi.org/10.1007/s00220-018-3193-x

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