Electronic Journal of Probability

A comparison of a nonlinear sigma model with general pinning and pinning at one point

Margherita Disertori, Franz Merkl, and Silke W.W. Rolles

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We study the nonlinear supersymmetric hyperbolic sigma model introduced by Zirnbauer in 1991. This model can be related to the mixing measure of a vertex-reinforced jump process. We prove that the two-point correlation function has a probabilistic interpretation in terms of connectivity in rooted random spanning forests. Using this interpretation, we dominate the two-point correlation function for general pinning, e.g. for uniform pinning, with the corresponding correlation function with pinning at one point. The result holds for a general finite graph, asymptotically as the strength of the pinning converges to zero. Specializing this to general ladder graphs, we deduce in the same asymptotic regime exponential decay of correlations for general pinning.

Article information

Electron. J. Probab., Volume 21 (2016), paper no. 27, 16 pp.

Received: 3 June 2015
Accepted: 16 March 2016
First available in Project Euclid: 8 April 2016

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Zentralblatt MATH identifier

Primary: 60G60: Random fields
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

nonlinear sigma model localization random spanning trees

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Disertori, Margherita; Merkl, Franz; Rolles, Silke W.W. A comparison of a nonlinear sigma model with general pinning and pinning at one point. Electron. J. Probab. 21 (2016), paper no. 27, 16 pp. doi:10.1214/16-EJP4340. https://projecteuclid.org/euclid.ejp/1460141798

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