The Annals of Applied Probability

Gibbs measures on permutations over one-dimensional discrete point sets

Marek Biskup and Thomas Richthammer

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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.

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Ann. Appl. Probab., Volume 25, Number 2 (2015), 898-929.

First available in Project Euclid: 19 February 2015

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Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 05A05: Permutations, words, matrices 82B10: Quantum equilibrium statistical mechanics (general)

Gibbs measures permutations extremal decomposition


Biskup, Marek; Richthammer, Thomas. Gibbs measures on permutations over one-dimensional discrete point sets. Ann. Appl. Probab. 25 (2015), no. 2, 898--929. doi:10.1214/14-AAP1013.

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