Critical two-point functions for long-range statistical-mechanical models in high dimensions

• Chen, Lung-Chi

• Sakai, Akira

Abstract

We consider long-range self-avoiding walk, percolation and the Ising model on Zd that are defined by power-law decaying pair potentials of the form D(x)≍|x|−d−α with α>0. The upper-critical dimension dc is 2(α∧2) for self-avoiding walk and the Ising model, and 3(α∧2) for percolation. Let α≠2 and assume certain heat-kernel bounds on the n-step distribution of the underlying random walk. We prove that, for d>dc (and the spread-out parameter sufficiently large), the critical two-point function Gpc(x) for each model is asymptotically C|x|α∧2−d, where the constant C∈(0,∞) is expressed in terms of the model-dependent lace-expansion coefficients and exhibits crossover between α<2 and α>2. We also provide a class of random walks that satisfy those heat-kernel bounds.

Journal

• The Annals of Probability

  The Annals of Probability 43 (2), 639-681, 2015-03

  Institute of Mathematical Statistics

Keywords

• Statistics, Probability and Uncertainty
• Statistics and Probability

Details

• CRID: 1050564288971392256
• NII Article ID: 120005530846
• DOI: 10.1214/13-aop843
• HANDLE: 2115/57803
• ISSN: 00911798
• Web Site: http://projecteuclid.org/download/pdfview_1/euclid.aop/1422885572
• Text Lang: en
• Article Type: journal article
• Data Source:
  • IRDB
  • Crossref
  • CiNii Articles
  • KAKEN