IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

An improved lower bound for the site percolation threshold of the hexagonal lattice

  • Oct. 16, 2020
  • 2:30 p.m.

Abstract

Percolation models are infinite random graph models which have applications to phase transitions and critical phenomena. In the site percolation model, each vertex in an infinite graph $G$ is retained independently with probability $p$ and deleted otherwise, while an edge is retained only if both of its endpoints are retained. Similarly, randomizing edges produces a bond percolation model.

The percolation threshold is the critical probability $p _ c(G)$ such that if $p > p _ c(G)$ there is positive probability that the random subgraph of retained elements has an infinite connected component, while the probability that all of its components are finite is one if $p < p _ c(G)$.

There are a few lattice graphs for which the site percolation threshold is exactly known. Rigorous bounds for unsolved lattices are very inaccurate. Current research is adapting the substitution method, which has been successful for computing relatively accurate bounds for some bond percolation thresholds, to site percolation models. The method has produced a new lower bound for the site percolation threshold of the hexagonal lattice, which was last improved in 1981.

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