Critical Parameters for Loop and Bernoulli Percolation

2019/08/27 by Peter Mühlbacher, Mühlbacher, Peter · 1 citation
Mathematics · Physics and Astronomy · #60K35 #82B26 #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.1908.10213

openalex publication_date 2019/08/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a class of random loop models (including the random interchange process) that are parametrised by a time parameter β≥ 0. Intuitively, larger β means more randomness. In particular, at β=0 we start with loops of length 1 and as β crosses a critical value β_c, infinite loops start to occur almost surely. Our random loop models admit a natural comparison to bond percolation with p=1-e^-β on the same graph to obtain a lower bound on β_c. For those graphs of diverging vertex degree where β_c and the critical parameter for percolation have been calculated explicitly, that inequality has been found to be an equality. In contrast, we show in this paper that for graphs of bounded degree the inequality is strict, i.e. we show existence of an interval of values of β where there are no infinite loops, but infinite percolation clusters almost surely.

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