The p-Spin Interaction Model with External Field

Abstract

This paper is devoted to a detailed study of a p-spins interaction model with external field, including some sharp bounds on the speed of self averaging of the overlap as well as a central limit theorem for its fluctuations, the thermodynamical limit for the free energy and the definition of an Almeida-Thouless type line. Those results show that the external field dominates the tendency to disorder induced by the increasing level of interaction between spins, and our system will share many of its features with the SK model, which is certainly not the case when the external magnetic field vanishes.Observe that we should write I = I(J), I c = I c (J), but we will omit this dependence for sake of readability. Using these sets, we defineProof: This proof is an extension of [9, Proposition 2.4.2], whose details are given for sake of completeness. We havewith κ n = n(n + 1). Note that, for any l ∈ {1, . . . , n},Thuswhere we recall that κ n = n(n + 1). This can be read asThe same kind of computations can be lead for A 1 , integrating first by parts with respect to the variables g I . In this case, we obtainSubstracting A 1 and A 2 , we get the desired result. As a consequence of the last proposition, we can bound ν k,t (f ) by ν k,1 (f ) as follows: 1 2 J∈Q p N,k z J ε l J , n l=1 ν k,t f ε l J ε n+1 J − n+1 l=1 ν k,t f ε l J ε n+1 J + n(n + 1)ν k,t f ε n+1 J ε n+2 J Now, from Lemma 8.4, we obtain easily u 2 N |Q p N,k | ≤ K N , that gives us the desired result.

Phase transition in the spiked random tensor with Rademacher prior

“…In the case of the Sherrington-Kirkpatrick model (p = 2), it was computed that β 2 = 1 (see Remark 2 in [19]) and the same results as Theorems 3 and 4 were obtained in Talagrand's book [50,Chapters 11 and 13]. As for p ≥ 3, Bardina, Márquez, Rovira, and Tindel [12] established Theorem 4 for some b β p as p increases. Our main contribution here is that the concentration of the overlap is valid up to the critical temperature.…”

The magnetization at high temperature for a p-spin interaction model with external field

“…For more information about the parameters β p and q p we refer the reader to Bardina et al (2004). As a consequence of Theorem 1.1, we have the following result:…”

“…In order to introduce our model we borrow the notations of Bardina et al (2004). The Hamiltonian of the p-spin interaction model with external field is defined by −H N,β,h (σ) = βu N (i 1 ,.…”

“…In mathematics, the p-spin interaction model is a natural generalization of the SK model (see Sherrington and Kirkpatrick (1975)). However, the mathematical papers devoted to this general kind of model are rare: see Talagrand (2000a) on low temperature regime; Bardina et al (2004) and Cadel et al (2004) on high temperature regime; and Bovier et al (2002) for some fluctuation results for the free energy.…”

Un théorème limite pour les covariances des spins dans le modèle de Sherrington–Kirkpatrick avec champ externe

“…Enfin, dans le travail [2], les auteursétudient le modèleà p spins avec champ externe et montrent, en utilisant ouétendant les méthodes de Talagrand, l'analogue des résultats obtenusà haute température pour p = 2 dans son livre [10].…”