Abstract.
We consider self-avoiding walk and percolation in ℤd, oriented percolation in ℤd×ℤ+, and the contact process in ℤd, with p D(·) being the coupling function whose range is proportional to L. For percolation, for example, each bond is independently 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 p c . We investigate the value of p c when d>6 for percolation and d>4 for the other models, and L≫1. We prove in a unified way that p c =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 random walk transition probability D. We also use this result to prove that p c =1+cL−d+O(L−d−1), where c is a model-dependent constant. Our proof is based on the lace expansion for each of these models.
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Hofstad, R., Sakai, A. Critical points for spread-out self-avoiding walk, percolation and the contact process above the upper critical dimensions. Probab. Theory Relat. Fields 132, 438–470 (2005). https://doi.org/10.1007/s00440-004-0405-4
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DOI: https://doi.org/10.1007/s00440-004-0405-4