GIBBS MEASURES ON PERMUTATIONS OVER ONE-DIMENSIONAL DISCRETE POINT SETS

We consider Gibbs distributions on permutations of a locally finite infinite set X subset of R, where a permutation sigma of X is assigned (formal) energy Sigma(x is an element of X) V(sigma(x) - x). This is motivated by Feynman's path representation of the quantum Bose gas; the choice X := Z and V(x) := alpha x(2) is of principal interest. Under suitable regularity conditions on the set X and the potential V, we establish existence and a full classification of the infinite-volume Gibbs measures for this problem, including a result on the number of infinite cycles of typical permutations. Unlike earlier results, our conclusions are not limited to small densities and/or high temperatures.