The band structure of a model of spatial random permutation

• 成果类型: Article
• 署名作者: Fyodorov, Yan V.; Muirhead, Stephen
• 署名单位: University of London; King's College London; University of Melbourne
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-020-01019-z
• 发表日期: 2021
• 页码: 543-587
• 关键词: transition

摘要

We study a random permutation of a lattice box in which each permutation is given a Boltzmann weight with energy equal to the total Euclidean displacement. Our main result establishes the band structure of the model as the box-size N tends to infinity and the inverse temperature beta tends to zero; in particular, we show that the mean displacement is of order min{1/beta,N}. In one dimension our results are more precise, specifying leading-order constants and giving bounds on the rates of convergence. Our proofs exploit a connection, via matrix permanents, between random permutations and Gaussian fields; although this connection is well-known in other settings, to the best of our knowledge its application to the study of random permutations is novel. As a byproduct of our analysis, we also provide asymptotics for the permanents of Kac-Murdock-Szego matrices.