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

• Chen, Lung-Chi ^[1] ; Sakai, Akira. ^[2]
  1. [1] Fu Jen Catholic University

     Fu Jen Catholic University

     Taiwán

     
  2. [2] Hokkaido University

     Hokkaido University

     Chūō-ku, Japón

     
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 43, Nº. 2, 2015, págs. 639-681
• Idioma: inglés
• DOI: 10.1214/13-AOP843
• Enlaces
  • Texto completo
• Resumen

  • 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.