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

Full-text: Open access

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1460141798

Digital Object Identifier
doi:10.1214/16-EJP4340

Mathematical Reviews number (MathSciNet)
MR3485369

Zentralblatt MATH identifier
1336.60097

Subjects
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.)

Keywords
nonlinear sigma model localization random spanning trees

Rights
Creative Commons Attribution 4.0 International License.

Citation

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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References

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