2019/04/24 by Thomas Mountford, Mountford, Thomas, Maria Eulália Vares +3
Mathematics · Physics and Astronomy · #60H15 #60K35 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1904.11030
openalex publication_date 2019/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article, we consider an anisotropic finite-range bond percolation model on ℤ2. On each horizontal layer \(x,i): x ∈ ℤ\ we have edges ⟨(x, i),(y, i)⟩ for 1 ≤ |x - y| ≤ N. There are also vertical edges connecting two nearest neighbor vertices on distinct layers ⟨(x, i), (x, i+1)⟩ for x, i ∈ℤ. On this graph we consider the following anisotropic independent percolation model: horizontal edges are open with probability 1/(2N), while vertical edges are open with probability ε to be suitably tuned as N grows to infinity. The main result tells that if ε=κN-2/5, we see a phase transition in κ: positive and finite constants C1, C2 exist so that there is no percolation if κ< C1 while percolation occurs for κ> C2. The question is motivated by a result on the analogous layered ferromagnetic Ising model at mean field critical temperature [J. Stat. Phys. (2015), 161, 91-123] where the authors showed the existence of multiple Gibbs measures for a fixed value of the vertical interaction and conjectured a change of behavior in κ when the vertical interaction suitably vanishes as κγb, where 1/γ is the range of the horizontal interaction. For the product percolation model we have a value of b that differs from what was conjectured in that paper. The proof relies on the analysis of the scaling limit of the critical branching random walk that dominates the growth process restricted to each horizontal layer and a careful analysis of the true horizontal growth process. This is inspired by works on the long range contact process [Probab. Th. Rel. Fields (1995), 102, 519-545]. A renormalization scheme is used for the percolative regime.