The band structure of a model of spatial random permutation

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.

Automated summary

The study examines a random permutation of a lattice box with a Boltzmann weight based on total Euclidean displacement. The main result establishes the band structure of the model as the box size and inverse temperature tend towards infinity and zero, respectively. The connection between random permutations and Gaussian fields via matrix permanents is utilized in the proofs, providing novel insights into the study of random permutations and yielding asymptotics for Kac-Murdock-Szego matrices.