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Exponential decay of the volume for Bernoulli percolation: a proof via stochastic comparison
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Let us consider subcritical Bernoulli percolation on a connected, transitive, infinite and locally finite graph. In this paper, we propose a new (and short) proof of the exponential decay property for the volume of clusters. We do not rely on differential inequalities and rather use stochastic comparison techniques, which are inspired by several works including the paper "An approximate zero-one law" written by Russo in the early eighties.
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Exponential decay for the random connection model using asymptotic transitivity
For any subcritical random connection model with integrable connection function, the cluster size distribution has an exponential tail.
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