×

Distances in the highly supercritical percolation cluster. (English) Zbl 1290.60099

Consider the grid that comprises all the points in \(\mathbb{Z}^{2}\). The so-called bond percolation is specified as follows. For each pair of points in \(\mathbb{Z}^{2}\) of distance \(1\) draw an edge with probability \(p\in (0,1)\) or leave the gap with probability \(1-p\); and do this independently from each pair of points. After doing this, there might be an infinite cluster of connected edges that passes through \((0,0)\), this event is denoted as \(\mathbf{0} \leftrightarrow \infty\). Now, let \(D(n,0)\) be the lower number of edges joining \((0,0)\) and \((n,0)\); if they are certainly connected it is denoted by \(\{\mathbf{0} \leftrightarrow (n,0)\}\). The authors prove that the quantity \(D(n,0)/n\) converges to the constant \(1+(1-p)/2+o(1-p)\), almost surely on the event \(\{\mathbf{0} \leftrightarrow \infty\}\), when \(n\to\infty\) but \(n\) such that \(\mathbf{0} \leftrightarrow (n,0)\). They base their proof on a connection with the so-called TASEP model, which is a stochastic model of particles that push each other in one direction. In turn, the authors show that there is a hidden TASEP inside the bond percolation, which helps them to analyze \(D(n,0)\).

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Durrett, R. (1984). Oriented percolation in two dimensions. Ann. Probab. 12 999-1040. · Zbl 0567.60095 · doi:10.1214/aop/1176993140
[2] Evans, M. R., Rajewsky, N. and Speer, E. R. (1999). Exact solution of a cellular automaton for traffic. J. Stat. Phys. 95 45-96. · Zbl 0964.82040 · doi:10.1023/A:1004521326456
[3] Garet, O. and Marchand, R. (2004). Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster. ESAIM Probab. Stat. 8 169-199 (electronic). · Zbl 1154.60356 · doi:10.1051/ps:2004009
[4] Gärtner, J. and Molchanov, S. A. (1990). Parabolic problems for the Anderson model. I. Intermittency and related topics. Comm. Math. Phys. 132 613-655. · Zbl 0711.60055 · doi:10.1007/BF02156540
[5] Grimmett, G. (1999). Percolation , 2nd ed. Grundlehren der Mathematischen Wissenschaften [ Fundamental Principles of Mathematical Sciences ] 321 . Springer, Berlin. · Zbl 0926.60004
[6] Hammersley, J. M. and Welsh, D. J. A. (1965). First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory. In Proc. Internat. Res. Semin. , Statist. Lab. , Univ. California , Berkeley , CA 61-110. Springer, New York. · Zbl 0143.40402
[7] Kesten, H. (1986). Aspects of first passage percolation. In École D’été de Probabilités de Saint-Flour , XIV- 1984. Lecture Notes in Math. 1180 125-264. Springer, Berlin. · Zbl 0602.60098
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.