Inequalities and Entanglements for Percolation and Random-Cluster Models

We discuss inequalities and applications for percolation and random-cluster models. The relevant areas of methodology concern the following two types of inequality: inequalities involving the probability of a general increasing event, and certain differential inequalities involving the percolation probability. We summarise three areas of application of such inequalities, namely strict inequality between the bond and site critical percolation probabilities of a general graph, the general study of entanglements in percolation, and strict inequalities for critical points of disordered random-cluster models.

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