Abstract
We use isoperimetric inequalities combined with a new technique to prove upper bounds for the site percolation threshold of plane graphs with given minimum degree conditions. In the process we prove tight new isoperimetric bounds for certain classes of hyperbolic graphs. This establishes the vertex isoperimetric constant for all triangular and square hyperbolic lattices, answering a question of Lyons and Peres. We prove that plane graphs of minimum degree at least 7 have site percolation threshold bounded away from 1/2, which was conjectured by Benjamini and Schramm, and make progress on a conjecture of Angel, Benjamini, and Horesh that the critical probability is at most 1/2 for plane triangulations of minimum degree 6. We prove additional bounds for stronger minimum degree conditions, and for graphs without triangular faces.
Highlights
In their highly influential paper [5], Benjamini and Schramm made several conjectures that generated a lot of interest among mathematicians and led to many beautiful mathematical results [2, 4, 6, 7, 12, 13], just to name a few
Despite the substantial amount of work, most of these conjectures are still open, while for a few of them, hardly anything is known. One of their conjectures states that ṗ c(G) < 1∕2 on any planar graph G of minimal degree at least 7; they conjecture that there are infinitely many infinite open clusters on the interval (ṗ c(G), 1 − ṗ c(G))
The connection between percolation thresholds and isoperimetric constants is well known, and in [5] it is proved that the site percolation threshold for a graph G is bounded above by (1 + ḣ (G))−1, where ḣ (G) is the vertex isoperimetric constant
Summary
In their highly influential paper [5], Benjamini and Schramm made several conjectures that generated a lot of interest among mathematicians and led to many beautiful mathematical results [2, 4, 6, 7, 12, 13], just to name a few. Benjamini, and Horesh considered isoperimetric inequalities for plane triangulations of minimum degree 6 in [1], and proved a discrete analogue of Weil’s theorem, showing that any such triangulation satisfies the same isoperimetric inequality as the Euclidean triangular lattice T6 They conjectured that T6 is extremal in other ways which might be expected to have connections with isoperimetric properties. They conjecture that the connective constant μ(T)—that is, the exponential growth rate of the number of self-avoiding walks of length n on T—is minimized by T6 among triangulations of minimum degree at least 6 They conjecture that percolation is hardest to achieve on T6 in the sense that both the critical probability for bond percolation pc(T) and the critical probability for site percolation ṗ c(T) are maximized by T6. See [9] for some further interesting properties and open problems regarding them
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