The Parisi formula for mixed $p$-spin models

The Aizenman-Sims-Starr scheme and Parisi formula for mixed p-spin spherical models

The Parisi formula for the free energy in the spherical models with mixed even p-spin interactions was proven in Michel Talagrand [16]. In this paper we study the general mixed p-spin spherical models including p-spin interactions for odd p. We establish the Aizenman-Sims-Starr scheme and from this together with many well-known results and Dmitry Panchenko’s recent proof on the Parisi ultrametr...

On Differentiability of the Parisi Formula

It was proved by Michel Talagrand in [10] that the Parisi formula for the free energy in the Sherrington-Kirkpatrick model is differentiable with respect to inverse temperature parameter. We present a simpler proof of this result by using approximate solutions in the Parisi formula and give one example of application of differentiability to prove non self-averaging of the overlap outside of the...

On Ward-takahashi Identities for the Parisi Spin Glass

The introduction of “small permutations” allows us to derive WardTakahashi identities for the spin-glass, in the Parisi limit of an infinite number of steps of replica symmetry breaking. The first identities express the emergence of a band of Goldstone modes . The next identities relate components of (the Replica Fourier Transformed) 3-point function to overlap derivatives of the 2-point functi...

Stability of the Parisi solution for Heisenberg spin glasses

Parisi function q(x) for spin glasses near Tc