We consider Gibbs distributions on permutations of a locally finite infinite
set XβR, where a permutation Ο of X is assigned
(formal) energy βxβXβV(Ο(x)βx). This is motivated by Feynman's
path representation of the quantum Bose gas; the choice X:=Z and
V(x):=Ξ±x2 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.Comment: Published in at http://dx.doi.org/10.1214/14-AAP1013 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org