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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 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.
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This paper has been withdrawn by the authors due to an error in the main theorem.
We consider the discrete Gaussian Free Field (DGFF) in scaled-up (square-lattic
The gamma kernels are a family of projection kernels $K^{(z,z)}=K^{(z,z)}(x,y)$ on a doubly infinite $1$-dimensional lattice. They are expressed through Eulers gamma function and depend on two continuous parameters $z,z$. The gamma kernels initially arose from a model of random partitions via a limit transition. On the other hand, these kernels are closely related to unitarizable representations of the Lie algebra $mathfrak{su}(1,1)$. Every gamma kernel $K^{(z,z)}$ serves as a correlation kernel for a determinantal measure $M^{(z,z)}$, which lives on the space of infinite point configurations on the lattice. We examine chains of kernels of the form $$ ldots, K^{(z-1,z-1)}, ; K^{(z,z)},; K^{(z+1,z+1)}, ldots, $$ and establish the following hierarchical relations inside any such chain: Given $(z,z)$, the kernel $K^{(z,z)}$ is a one-dimensional perturbation of (a twisting of) the kernel $K^{(z+1,z+1)}$, and the one-point Palm distributions for the measure $M^{(z,z)}$ are absolutely continuous with respect to $M^{(z+1,z+1)}$. We also explicitly compute the corresponding Radon-Nikodym derivatives and show that they are given by certain normalized multiplicative functionals.
We show that nontrivial bi-infinite polymer Gibbs measures do not exist in typical environments in the inverse-gamma (or log-gamma) directed polymer model on the planar square lattice. The precise technical result is that, except for measures supported on straight-line paths, such Gibbs measures do not exist in almost every environment when the weights are independent and identically distributed inverse-gamma random variables. The proof proceeds by showing that when two endpoints of a point-to-point polymer distribution are taken to infinity in opposite directions but not parallel to lattice directions, the midpoint of the polymer path escapes. The proof is based on couplings, planar comparison arguments, and a recently discovered joint distribution of Busemann functions.
We define a potential-weighted connective constant that measures the effective strength of a repulsive pair potential of a Gibbs point process modulated by the geometry of the underlying space. We then show that this definition leads to improved bounds for Gibbs uniqueness for all non-trivial repulsive pair potentials on $mathbb R^d$ and other metric measure spaces. We do this by constructing a tree-branching collection of densities associated to the point process that captures the interplay between the potential and the geometry of the space. When the activity is small as a function of the potential-weighted connective constant this object exhibits an infinite volume uniqueness property. On the other hand, we show that our uniqueness bound can be tight for certain spaces: the same infinite volume object exhibits non-uniqueness for activities above our bound in the case when the underlying space has the geometry of a tree.
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