Electronic Communications in Probability

A note on Talagrand's positivity principle

Dmitriy Panchenko

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Abstract

Talagrand's positivity principle states that one can slightly perturb a Hamiltonian in the Sherrington-Kirkpatrick model in such a way that the overlap of two configurations under the perturbed Gibbs' measure will become typically nonnegative. In this note we observe that abstracting from the setting of the SK model only improves the result and does not require any modifications in Talagrand's argument. In this version, for example, positivity principle immediately applies to the setting of replica symmetry breaking interpolation. Also, abstracting from the SK model improves the conditions in the Ghirlanda-Guerra identities and as a consequence results in a perturbation of smaller order necessary to ensure positivity of the overlap.

Article information

Source
Electron. Commun. Probab., Volume 12 (2007), paper no. 38, 401-410.

Dates
Accepted: 21 October 2007
First available in Project Euclid: 6 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465224981

Digital Object Identifier
doi:10.1214/ECP.v12-1326

Mathematical Reviews number (MathSciNet)
MR2350577

Zentralblatt MATH identifier
1140.60355

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Keywords
Talagrand's positivity principle Ghirlanda-Guerra identities

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Panchenko, Dmitriy. A note on Talagrand's positivity principle. Electron. Commun. Probab. 12 (2007), paper no. 38, 401--410. doi:10.1214/ECP.v12-1326. https://projecteuclid.org/euclid.ecp/1465224981


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References

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