GIBBS MEASURES ON PERMUTATIONS OVER ONE-DIMENSIONAL DISCRETE POINT SETS

• 成果类型: Article
• 署名作者: Biskup, Marek; Richthammer, Thomas
• 署名单位: University of California System; University of California Los Angeles; University of South Bohemia Ceske Budejovice; University of Hildesheim; University of Hildesheim
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/14-AAP1013
• 发表日期: 2015
• 页码: 898-929
• 关键词: percolation transition

摘要

We consider Gibbs distributions on permutations of a locally finite infinite set X subset of R, where a permutation sigma of X is assigned (formal) energy Sigma(x is an element of X) V(sigma(x) - x). This is motivated by Feynman's path representation of the quantum Bose gas; the choice X := Z and V(x) := alpha x(2) is of principal interest. Under suitable regularity conditions on the set X and the potential V, we establish existence and a full classification of the infinite-volume Gibbs measures for this problem, including a result on the number of infinite cycles of typical permutations. Unlike earlier results, our conclusions are not limited to small densities and/or high temperatures.