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A note on Talagrand’s positivity principle. (English) Zbl 1140.60355
Summary: 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 St. Ghirlanda and F. Guerra [J. Phys. A, Math. Gen. 31, No. 46, 9149–9155 (1998; Zbl 0953.82037)] identities and as a consequence results in a perturbation of smaller order necessary to ensure positivity of the overlap.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
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