Ising Percolation on Nonamenable Planar Graphs

2020/06/13 by Li, Zhongyang
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2006.09218

Abstract

We study infinite ``+'' or ``-'' clusters for an Ising model on an connected, transitive, non-amenable, planar, one-ended graph G with finite vertex degree. If the critical percolation probability p_c^site for the i.i.d.~Bernoulli site percolation on G is less than (1)/(2), we find an explicit region for the coupling constant of the Ising model such that there are infinitely many infinite ``+''-clusters and infinitely many infinite ``-''-clusters, while the random cluster representation of the Ising model has no infinite 1-clusters. If p_c^site>(1)/(2), we obtain a lower bound for the critical probability in the random cluster representation of the Ising model in terms of p_c^site.

Related

• Constrained percolation, Ising model and XOR Ising model on planar lattices
• Counting independent sets in percolated graphs via the Ising model
• Absence of infinite cluster for critical Bernoulli percolation on slabs
• Percolation in the two-dimensional Ising model
• The high temperature Ising model on the triangular lattice is a critical percolation model