Percolation and local isoperimetric inequalities (Q737324)

The author considers the site percolation model on a connected graph \(G\) of polynomial growth, which is such that each vertex of \(G\) is kept with probability \(p\) and removed otherwise. The author establishes a partial answer to a question posed by \textit{I. Benjamini} and \textit{O. Schramm} [Electron. Commun. Probab. 1, 71--82 (1996; Zbl 0890.60091)] on the existence of a non-trivial phase transition for the above model. To formulate the main result of the paper, the following definition of the local isoperimetric inequality is introduced. A graph \(G\) is said to satisfy the local isoperimetric inequality if for any sets \(A\subseteq B(x,r):=B\) such that \(| A| \leq | B|/2\) we have \[ |\partial_B A|\geq c| A|^{\frac{d}{d-1}}. \] It is proved that if a graph \(G\) obeys the above inequality with \(d>1\), then the critical probability for the percolation model on \(G\) is \(<1\).