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arXiv · 2006.09218

Ising Percolation on Nonamenable Planar Graphs

Zhongyang Li
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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 $\frac{1}{2}$, 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 $p_c^{site}>\frac{1}{2}$, we obtain a lower bound for the critical probability in the random cluster representation of the Ising model in terms of $p_c^{site}$.

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Zhongyang Li. 2020-06-13. Ising Percolation on Nonamenable Planar Graphs. https://arxiv.org/abs/2006.09218

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