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arXiv · 0903.1501

Sharp thresholds for the random-cluster and Ising models

Benjamin Graham·Geoffrey Grimmett
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Abstract

A sharp-threshold theorem is proved for box-crossing probabilities on the square lattice. The models in question are the random-cluster model near the self-dual point $p_{\mathrm {sd}}(q)=\sqrt{q}/(1+\sqrt{q})$, the Ising model with external field, and the colored random-cluster model. The principal technique is an extension of the influence theorem for monotonic probability measures applied to increasing events with no assumption of symmetry.

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Benjamin Graham, Geoffrey Grimmett. 2009-03-09. Sharp thresholds for the random-cluster and Ising models. https://doi.org/10.1214/10-aap693

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