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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.