Percolation on random 2-lifts

2025/06/02 by Paul Drouvillé, Drouvillé, Paul
Mathematics · Physics and Astronomy · #60K35 #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2506.01612

openalex publication_date 2025/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a graph G, we consider a model for a random cover of G by taking two parallel copies of G and crossing every pair of parallel edges randomly with probability q independently of each other. The resulting graph G_q, is a random 2-lift of G that may not be transitive but still probabilistically exhibit many properties of transitive graphs. Studying percolation in this context can help us test the reliability and robustness of our proofs methods in percolation theory. Our three main results on this model are the continuity of the critical parameter p_c(G_q), for q∈(0,1), the strict monotonicity p_c(G_q)< p_c(G) and the exponential decay of the cluster size in the subcritical regime at q=1/2.

Citations

• Exponential decay of the volume for Bernoulli percolation: a proof via stochastic comparison
• Strict monotonicity of percolation thresholds under covering maps
• Sharp phase transition for the random-cluster and Potts models via\n decision trees
• A new proof of the sharpness of the phase transition for Bernoulli percolation on \mathbb Z^d
• Essential enhancements revisited
• Interlacing Families I: Bipartite Ramanujan Graphs of All Degrees
• The spectrum of random lifts
• Is the critical percolation probability local?

Related

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• Percolations on random maps I: half-plane models
• Mean-field conditions for percolation on finite graphs
• Bootstrap percolation on the high-dimensional Hamming graph
• The critical percolation window in growing random graphs