Upper Bound for Critical Probability of Site Percolation on Triangular Lattice

TL;DR

This paper discusses the critical probability for site percolation on a triangular lattice, highlighting known exact and approximate values and their implications for infinite open vertex sets.

Contribution

It provides an upper bound for the critical probability in site percolation on the triangular lattice, building on Smirnov's findings.

Findings

Critical p for triangular lattice is 0.5 according to Smirnov.

Existence of an infinite open vertex set at p=0.3535.

Upper bound established for the critical probability.

Abstract

In site percolation, vertices (sites) of a graph are open with probability p, and there is critical p, for which open vertices form an open path the long way across a graph, so a vertex at the origin is a part of an infinite connected open vertex set. Smirnov found that for triangular lattice critical p is 0.5, but there is the traversal, from the origin upwards, so that an infinite connected open vertex set exists for critical p=0.3535.