The Uniform Even Subgraph and Its Connection to Phase Transitions of Graphical Representations of the Ising Model

2023/06/08 by Ulrik Thinggaard Hansen, Hansen, Ulrik Thinggaard, Boris Kjær +3 · 2 citations
Mathematics · Physics and Astronomy · #05C80 #60K35 #82B05 82B20 #82B26 #82B43 #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.2306.05130

openalex publication_date 2023/06/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The uniform even subgraph is intimately related to the Ising model, the random-cluster model, the random current model, and the loop O(1) model. In this paper, we first prove that the uniform even subgraph of Z^d percolates for d ≥ 2 using its characterisation as the Haar measure on the group of even graphs. We then tighten the result by showing that the loop O(1) model on Z^d percolates for d ≥ 2 for edge-weights x lying in some interval (1-ε,1]. Finally, our main theorem is that the loop O(1) model and random current models corresponding to a supercritical Ising model are always at least critical, in the sense that their two-point correlation functions decay at most polynomially and the expected cluster sizes are infinite.

Cited by

• A simple proof of exponential decay for the near-critical planar Ising model
• A General Coupling for Ising Models and Beyond

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