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 \(D(n,0)\).


60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
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