open access

https://hdl.handle.net/2115/44916

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

英語

2005

Springer

Probability Theory and Related Fields

132(3)

438-470

We consider self-avoiding walk and percolation in Zd , oriented percolation in Zd × Z+, and the contact process in Zd , with pD( · ) being the coupling function whose range is proportional to L. For percolation, for example, each bond is independently occupied with probabilitypD(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 random walk transition probability D. We also use this result to prove that pc = 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.

学術雑誌論文

Accepted Manuscript

"The original publication is available at www.springerlink.com"

0178-8051

AA10531847