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2020 IPS Conference
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Corporate Members

- IPS Conference 2018

We consider a cubic lattice of *N=L*^{3} sites. A random walk (RW) starts at a random position and performs *uN* steps. The boundaries of the cube are considered to be periodic. Consequently, the RW keeps visiting sites within the cube, some of them repeatedly. We consider percolation of the vacant sites, not visited by the RW. As *u* increases, the fraction of vacant sites *p* decreases. This problem was originally considered by Abete *et al.* [1] and denoted as `pacman percolation.' Later, mathematical aspects of this (`interlacement') problem were considered more rigorously [2], and the existence of critical value *u _{c}* below which the vacant sites percolate was proven. The vacant sites are strongly correlated, and significant finite size effects are expected. Our study considers significantly larger latices (up to

[1] T. Abete, A. de Candia, D. Lairez, and A. Coniglio, Phys. Rev. Lett. 93, 228301 (2004).

[2] A.-S. Sznitman, The Annals of Probability 36, 1 (2008); V. Sidoravicius and A.-S. Sznitman, Commun. Pure Appl. Math. 62, 0831 (2009).

[3] E. W. Montroll and G. H. Weiss, J. Math. Phys. 6, 167 (1965).

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