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Thermodynamic limit for the Mallows model on Sn

J. Math. Phys. 50, 095208 (2009); doi:10.1063/1.3156746

Published 10 July 2009

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Shannon Starr
Department of Mathematics, Hylan Building, University of Rochester, Rochester, New York 14627, USA
The Mallows model on Sn is a probability distribution on permutations, qd(pi,e)/Pn(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<=i<j<=n, but pii>pij. Analyzing the normalization Pn(q), Diaconis and Ram calculated the mean and variance of d(pi,e) in the Mallows model, which suggests that the appropriate n-->[infinity] limit has qn scaling as 1−beta/n. We calculate the distribution of the empirical measure in this limit, u(x,y)dxdy=limn-->[infinity](1/n)[summation]<sub>i = 1</sub><sup>n</sup>delta(i,pii). Treating it as a mean-field problem, analogous to the Curie–Weiss model, the self-consistent mean-field equations are ([partial-derivative]2/[partial-derivative]x[partial-derivative]y)ln u(x,y)=2betau(x,y), which is an integrable partial differential equation, 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 [script U]q([fraktur s][fraktur l]2)-symmetric XXZ ferromagnet. ©2009 American Institute of Physics
History: Received 4 April 2009; accepted 28 May 2009; published 10 July 2009
Permalink: http://link.aip.org/link/?JMAPAQ/50/095208/1
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KEYWORDS and PACS

Keywords
PACS
  • 05.70.-a
    Thermodynamics
  • 02.60.Nm
    Integral and integrodifferential equations
  • 02.60.Lj
    Ordinary and partial differential equations; boundary value problems
  • 02.50.Cw
    Probability theory
  • 02.30.Jr
    Partial differential equations
  • YEAR: 2009

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PUBLICATION DATA

ISSN:
0022-2488 (print)   1089-7658 (online)
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