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Title: The covariance matrix of the Potts model: A random cluster analysis

Abstract

We consider the covariance matrix, G{sup mn} = q{sup 2}<{delta}({sigma}{sub x},m); {delta}({sigma}{sub y},n)>, of the d-dimensional q-states Potts model, rewriting it in the random cluster representation of Fortuin and Kasteleyn. In many of the q ordered phases, we identify the eigenvalues of this matrix both in terms of representations of the unbroken symmetry group of the model and in terms of random cluster connectivities and covariances, thereby attributing algebraic significance to these stochastic geometric quantities. We also show that the correlation length and the correlation length corresponding to the decay rate of one on the eigenvalues in the same as the inverse decay rate of the diameter of finite clusters. For dimension of d=2, we show that this correlation length and the correlation length of two-point function with free boundary conditions at the corresponding dual temperature are equal up to a factor of two. For systems with first-order transitions, this relation helps to resolve certain inconsistencies between recent exact and numerical work on correlation lengths at the self-dual point {beta}{sub o}. For systems with second order transitions, this relation implies the equality of the correlation length exponents from above below threshold, as well as an amplitude ratio of two. Inmore » the course of proving the above results, we establish several properties of independent interest, including left continuity of the inverse correlation length with free boundary conditions and upper semicontinuity of the decay rate for finite clusters in all dimensions, and left continuity of the two-dimensional free boundary condition percolation probability at {beta}{sub o}. We also introduce DLR equations for the random cluster model and use them to establish ergodicity of the free measure. In order to prove these results, we introduce a new class of events which we call decoupling events and two inequalities for these events.« less

Authors:
 [1];  [2]
  1. Charles Univ., Prague (Czechoslovakia)
  2. UCLA, Los Angeles, CA (United States)
Publication Date:
Sponsoring Org.:
USDOE
OSTI Identifier:
468313
Resource Type:
Journal Article
Journal Name:
Journal of Statistical Physics
Additional Journal Information:
Journal Volume: 82; Journal Issue: 5-6; Other Information: PBD: Mar 1996
Country of Publication:
United States
Language:
English
Subject:
66 PHYSICS; 99 MATHEMATICS, COMPUTERS, INFORMATION SCIENCE, MANAGEMENT, LAW, MISCELLANEOUS; STATISTICAL MODELS; MATRICES; MAGNETISM; CORRELATIONS; DATA COVARIANCES; BOUNDARY CONDITIONS

Citation Formats

Borgs, C, Freie Universitaet, Berlin, and Chayes, J T. The covariance matrix of the Potts model: A random cluster analysis. United States: N. p., 1996. Web. doi:10.1007/BF02183383.
Borgs, C, Freie Universitaet, Berlin, & Chayes, J T. The covariance matrix of the Potts model: A random cluster analysis. United States. https://doi.org/10.1007/BF02183383
Borgs, C, Freie Universitaet, Berlin, and Chayes, J T. Fri . "The covariance matrix of the Potts model: A random cluster analysis". United States. https://doi.org/10.1007/BF02183383.
@article{osti_468313,
title = {The covariance matrix of the Potts model: A random cluster analysis},
author = {Borgs, C and Freie Universitaet, Berlin and Chayes, J T},
abstractNote = {We consider the covariance matrix, G{sup mn} = q{sup 2}<{delta}({sigma}{sub x},m); {delta}({sigma}{sub y},n)>, of the d-dimensional q-states Potts model, rewriting it in the random cluster representation of Fortuin and Kasteleyn. In many of the q ordered phases, we identify the eigenvalues of this matrix both in terms of representations of the unbroken symmetry group of the model and in terms of random cluster connectivities and covariances, thereby attributing algebraic significance to these stochastic geometric quantities. We also show that the correlation length and the correlation length corresponding to the decay rate of one on the eigenvalues in the same as the inverse decay rate of the diameter of finite clusters. For dimension of d=2, we show that this correlation length and the correlation length of two-point function with free boundary conditions at the corresponding dual temperature are equal up to a factor of two. For systems with first-order transitions, this relation helps to resolve certain inconsistencies between recent exact and numerical work on correlation lengths at the self-dual point {beta}{sub o}. For systems with second order transitions, this relation implies the equality of the correlation length exponents from above below threshold, as well as an amplitude ratio of two. In the course of proving the above results, we establish several properties of independent interest, including left continuity of the inverse correlation length with free boundary conditions and upper semicontinuity of the decay rate for finite clusters in all dimensions, and left continuity of the two-dimensional free boundary condition percolation probability at {beta}{sub o}. We also introduce DLR equations for the random cluster model and use them to establish ergodicity of the free measure. In order to prove these results, we introduce a new class of events which we call decoupling events and two inequalities for these events.},
doi = {10.1007/BF02183383},
url = {https://www.osti.gov/biblio/468313}, journal = {Journal of Statistical Physics},
number = 5-6,
volume = 82,
place = {United States},
year = {1996},
month = {3}
}