The Aizenman-Sims-Starr scheme and Parisi formula for mixed \(p\)-spin spherical models. (English) Zbl 1288.60127

Within the Sherrington-Kirkpatrick model of Ising spin glass, the variational Parisi formula for its free energy has been derived, based on the hierarchical replica symmetry breaking scheme. Despite its mathematical beauty, it is hardly amenable to any direct computation. Therefore, a closely related spherical model has been introduced that retains basic feature of the Ising case. The Parisi formula analogue is known to be valid there and a more explicit representation for the free energy is possible in this framework. By encompassing a series of previous results, the Parisi formula for the Ising spin glass model with general \(p\)-spin interaction has been proved by D. Panchenko [Ann. Math. (2) 177, No. 1, 383–393 (2013; Zbl 1270.60060); Ann. Probab. 41, No. 3A, 1315–1361 (2013; Zbl 1281.60081)]. The main goal of the present author is to establish an extension of the Aizenman-Simms-Starr variational scheme and to prove next the Parisi formula in the spherical model including \(p\)-spin interactions for odd \(p\). The previous seminal paper by M. Talagrand [Probab. Theory Relat. Fields 134, No. 3, 339–382 (2006; Zbl 1130.82019)] on the spherical case addressed mixed even \(p\)-spin interactions only.


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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