2015/04/29 by Yinshan Chang, Chang, Yinshan · 1 citation
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR
paper · pdf · doi:10.48550/arxiv.1504.07906
arxiv created 2015/04/29 · arxiv updated 2015/04/30
In this paper, we are interested in the loop cluster model on ℤd for d≥ 3. It is a long range model with two parameters α and κ, where the non-negative parameter α measures the amount of loops, and κ plays the role of killing on vertices penalizing (κ≥ 0) or favoring (κ<0) appearance of large loops. We consider the truncated loop cluster model formed by the Poisson point process Lα,≤ m, which is the restriction of Lα on loops with at most m jumps. We prove the existence of percolation in a 2-dimensional slab for the truncated loop model Lα,≤ m as long as the intensity parameter α is strictly above the critical threshold of the non-truncated loop model and m is large enough. We apply this result to prove the exponential decay of one arm connectivity for the finite cluster at 0 for the whole supercritical regime of the non-truncated loop model. For κ=0, this loop percolation model provides an example in which we have different behaviors of finite clusters in sub-critical and super-critical regimes. Also, we deduce the strict increase of the critical curve α→κc(α) for α≥αc, where αc is the critical value when κ=0. In the end, we prove that ∀α>αc large balls in the infinite cluster are finally very regular in the sense of \citeSapozhnikov2014, which implies that large balls are finally very good in the sense of \citeBarlowMR2094438. By \citeBarlowMR2094438 and \citeBarlowHamblyMR2471657, we have Harnack's inequality and Gaussian type estimate for simple random walks on the infinite cluster for all α>αc.