  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 (KMS) matrices.
