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Percolation with Constant Freezing

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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 \(\mathbb {Z}^d\), and many open questions are discussed.

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Acknowledgments

I would like to thank Nathanaël Berestycki for suggesting this problem to me, for helpful discussion and for careful reading of various drafts of this paper. I am also grateful to Geoffrey Grimmett for suggesting I use ergodic decomposition in a similar way to [9]—see Lemma 2.15, and to the anonymous referee for their constructive comments. This work has been supported by the UK Engineering and Physical Sciences Research Council (EPSRC) Grant EP/H023348/1.

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  1. Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK

    Edward Mottram

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  1. Edward Mottram
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Correspondence to Edward Mottram.

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Mottram, E. Percolation with Constant Freezing. J Stat Phys 155, 932–965 (2014). https://doi.org/10.1007/s10955-014-0985-z

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  • DOI: https://doi.org/10.1007/s10955-014-0985-z

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