Percolation of the Vacant Sites in the Interlacement Problem

  Yacov Kantor [1]  ,  Mehran Kardar [2]  
[1] Tel Aviv University
[2] Massachusetts Institute of Technology

We consider a cubic lattice of N=L3 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 uc 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 L=512) than in the earlier work, and besides confirming the original predictions of [1], we study additional properties of the problem. We show that the fraction of the vacant sites p=exp(-0.6595 u), where the numerical prefactor is consistent with the results obtained many years ago in [3]. We find that the critical value uc=3.15±0.01, and, correspondingly, the critical pc=0.125±0.001. By measuring the width of the percolation transition we find the correlation length exponent ν≈2.10, which is close to ν=2 predicted in [1]. Percolation probability Πc exactly at pc when the size of system diverges is expected to be a universal quantity. We find Πc=0.04±0.01, slightly larger than obtained in [1] but still consistent with it. This number is significantly smaller than what is expected in the regular (Bernoulli) percolation.

[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).