The band structure of a model of spatial random permutation (Q2663398)

A spatial random permutation (SRP) is a probability measure on a set of permutations which is biased towards the identity in some underlying geometry. In the Mallows model of SRP each permutation \(\pi\) of an \(N\)-element set is assigned weight \(P(\pi)\propto q^{\mathrm{inv}(\pi)}\), where \(q\in (0,1]\) is a parameter and \(\mathrm{inv}(\pi):=|\{(s,t): s<t, \pi(s)>\pi(t)\}|\) is the inversion count of \(\pi\). In the case \(q=1\), the Mallows model reduces to the classical model of random uniform permutation. A more general class of SRPs are the Boltzmann SRPs whose permutation weights are given by \(P(\pi)\propto e^{-\beta H(\pi)}\), where \(\beta\ge 0\) is an inverse temperature parameter and \(H(\pi)\) is an energy function that depends on the distance from the identity in an underlying geometry. The case \(\beta=0\) corresponds to the uniform random permutation. Let \(\mathcal{T}\) be a finite set of points in \(\mathbb{R}^d\) and let \(V:\mathbb{R}^d\to\mathbb{R}^+\) be a potential function. To each bijection \(\pi\) on the set \(\mathcal{T}\) one can associate an energy function \(H(\pi):=\sum_{x\in\mathcal{T}} V(x-\pi(x)))\). The class of random permutation models with an energy function of this form is referred to random Euclidean bijections. Natural choices for the potential \(V\) include \(V(x)=|x|\) and \(V(x)=|x|^2\), where \(|\cdot|\) denotes the standard Euclidean distance on \(\mathbb{R}^d\). In this paper, the authors study random permutations equipped with the Boltzmann weights whose energy equals the total Euclidean displacement. From the authors' abstract: ``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 rates of convergence. Our proofs exploit a connection, via matrix permanents, between random permutatons 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-Szegö matrices.''