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–Szegő matrices.

我们研究了晶格盒的随机排列，其中每个排列都赋予Boltzmann权重，其能量等于总的欧几里得位移。我们的主要结果建立了模型的能带结构，因为盒子大小N趋于无穷大，逆温度 \（\ beta \）趋于零；特别是，我们证明平均位移为\（\ min \ {1 / \ beta，N \} \）。在一个维度上，我们的结果更加精确，指定了前导常数，并给出了收敛速度的界限。我们的证明通过矩阵永久性利用随机置换和高斯场之间的联系。尽管这种联系在其他情况下是众所周知的，但据我们所知，它在随机排列研究中的应用是新颖的。作为我们分析的副产品，我们还为Kac-Murdock-Szegő矩阵的永久性提供了渐近性。