Percolation with constant freezing

TL;DR

This paper introduces a new percolation model with constant cluster freezing rate, demonstrating the existence of an infinite volume process on amenable graphs and revealing unique critical behaviors and algebraic decay properties.

Contribution

It constructs the infinite volume process for percolation with constant freezing on amenable graphs and analyzes its critical properties and cluster size distributions.

Findings

Infinite volume process exists on amenable graphs.

Cluster size distribution exhibits algebraic tail decay.

No lower critical dimension suggested by degree-dependent exponents.

Abstract

We introduce and study a model of percolation with constant freezing (PCF) where edges open at constant rate 1, and clusters freeze at rate \alpha independently of their size. Our main result is that the infinite volume process can be constructed on any amenable vertex transitive graph. This is in sharp contrast to models of percolation with freezing previously introduced, where the limit is known not to exist. Our interest is in the study of the percolative properties of the final configuration as a function of \alpha. We also obtain more precise results in the case of trees. Surprisingly the algebraic exponent for the cluster size depends on the degree, suggesting that there is no lower critical dimension for the model. Moreover, even for \alpha<\alpha_c, it is shown that finite clusters have algebraic tail decay, which is a signature of self organised criticality. Partial results are obtained on Z^d, and many open questions are discussed.