We consider Gibbs distributions on permutations of a locally finite infinite set $Xsubsetmathbb{R}$, where a permutation $sigma$ of $X$ is assigned (formal) energy $sum_{xin X}V(sigma(x)-x)$. This is motivated by Feynmans path representation of the q
uantum 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.
We consider gradient fields $(phi_x:xin mathbb{Z}^d)$ whose law takes the Gibbs--Boltzmann form $Z^{-1}exp{-sum_{< x,y>}V(phi_y-phi_x)}$, where the sum runs over nearest neighbors. We assume that the potential $V$ admits the representation [V(eta):=-
logintvarrho({d}kappa)expbiggl[-{1/2}kappaet a^2biggr],] where $varrho$ is a positive measure with compact support in $(0,infty)$. Hence, the potential $V$ is symmetric, but nonconvex in general. While for strictly convex $V$s, the translation-invariant, ergodic gradient Gibbs measures are completely characterized by their tilt, a nonconvex potential as above may lead to several ergodic gradient Gibbs measures with zero tilt. Still, every ergodic, zero-tilt gradient Gibbs measure for the potential $V$ above scales to a Gaussian free field.
