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Thermodynamic limit for the Mallows model on \(S_n\). (English) Zbl 1241.82039
Summary: The Mallows model on \(S_n\) is a probability distribution on permutations, \(q^{d(\pi,e)}/P_n(q)\), where \(d(\pi,e)\) is the distance between \(\pi\) and the identity element, relative to the Coxeter generators. Equivalently, it is the number of inversions: pairs \((i,j)\) where \(1\leq i<j\leq n\), but \(\pi_i>\pi_j\). Analyzing the normalization \(P_n(q)\), P. Diaconis and A. Ram [Mich. Math. J. 48, Spec. Vol., 157–190 (2000; Zbl 0998.60069)] calculated the mean and variance of \(d(\pi,e)\) in the Mallows model, which suggests the appropriate \(n \to \infty\) limit has \(q_n\) scaling as \(1-\beta/n\). We calculate the distribution of the empirical measure in this limit, \[ u(x,y) \,dx \,dy = \lim_{n \to \infty} \frac{1}{n} \sum_{i=1}^{n} \delta_{(i,\pi_i)}. \] Treating it as a mean-field problem, analogous to the Curie-Weiss model, the self-consistent mean-field equations are \(\frac{\partial^2}{\partial x \partial y} \ln u(x,y) = 2 \beta u(x,y)\), which is an integrable PDE, known as the hyperbolic Liouville equation. The explicit solution also gives a new proof of formulas for the blocking measures in the weakly asymmetric exclusion process, and the ground state of the \(\mathcal{U}_q(\mathfrak{sl}_2)\)-symmetric \(XXZ\) ferromagnet.
©2009 American Institute of Physics

MSC:
82B31 Stochastic methods applied to problems in equilibrium statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60K40 Other physical applications of random processes
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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