Gibbs measures on permutations over one-dimensional discrete point sets

Abstract

We consider Gibbs distributions on permutations of a locally finite infinite set $X\subset \mathbb{R}$, where a permutation $\sigma$ of $X$ is assigned (formal) energy ${\sum }_{x\in X}V(\sigma (x)-x)$. This is motivated by Feynman's path representation of the quantum Bose gas; the choice $X:=\mathbb{Z}$ 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.

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)

References in corpus (5)

• Spatial random permutations and infinite cycles
• Lattice permutations and Poisson-Dirichlet distribution of cycle lengths
• The relation between Feynman cycles and off-diagonal long-range order
• Infinite cycles in the random stirring model on trees
• Random permutations of a regular lattice

Cited by in corpus (1)

• Regenerative random permutations of integers