Critical points for spread-out self-avoiding walk, percolation and the contact process above the upper critical dimensions

2004/02/04 by Remco van der Hofstad, Akira Sakai, van der Hofstad, Remco +1 · 2 citations
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #math-ph #math.MP #math.PR

paper · pdf · doi:10.48550/arxiv.math/0402050

22 pages, no figures

arxiv created 2004/09/24 · arxiv updated 2009/12/01

Abstract

We consider self-avoiding walk and percolation in \Zd, oriented percolation in \Zd×\Zp, and the contact process in \Zd, with p D(⋅) being the coupling function whose range is denoted by L<∞. For percolation, for example, each bond \x,y\ is occupied with probability p D(y-x). The above models are known to exhibit a phase transition when the parameter p varies around a model-dependent critical point \pc. We investigate the value of \pc when d>6 for percolation and d>4 for the other models, and L≫1. We prove in a unified way that \pc=1+C(D)+O(L^-2d), where the universal term 1 is the mean-field critical value, and the model-dependent term C(D)=O(L^-d) is written explicitly in terms of the function D. Our proof is based on the lace expansion for each of these models.

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• Estimating the critical threshold for stretched-out hyperbolic random connection models
• Asymptotic expansion of the critical point for oriented percolation in high dimensions

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