Fluctuations of the partition function in the generalized random energy model with external field

Abstract

We study Derrida's generalized random energy model ͑GREM͒ in the presence of uniform external field. We compute the fluctuations of the ground state and of the partition function in the thermodynamic limit for all admissible values of parameters. We find that the fluctuations are described by a hierarchical structure which is obtained by a certain coarse graining of the initial hierarchical structure of the GREM with external field. We provide an explicit formula for the free energy of the model. We also derive some large deviation results providing an expression for the free energy in a class of models with Gaussian Hamiltonians and external field. Finally, we prove that the coarse-grained parts of the system emerging in the thermodynamic limit tend to have a certain optimal magnetization, as prescribed by the strength of the external field and by parameters of the GREM.

On the energy landscape of the mixed even p-spin model

“…Recall the notations used in the proof of Lemma 1. Let u γ 0 , u γ θ , u γ 1 be the corresponding maximizers of (13) generated by (10). From Lemma 1,…”

The de Almeida–Thouless Line in Hierarchical Quantum Spin Glasses

“…3, is a generalization of Theorem 1.4 in [22], which addresses the case without a longitudinal field, h = 0. In the classical case without transversal magnetic field, b = 0, it generalizes the results of [7], which covers the case that h is constant, and of [4,5], which treats the special case of a REM or two-level GREM in a random magnetic field.…”

“…therein). Unlike for the SK-model, implementing the longitudinal field naively in GREM models causes the frozen phase to expand [4,5,7]. Derrida and Gardner [14] therefore suggested a hierarchical implementation of the longitudinal magnetic field, which then leads again to a destabilization of the frozen phase.…”

“…Glass behavior occurs if the inverse temperature β ≥ β c (x) for at least one x ∈ [0, 1]. From (8) and the monotonicity of ϕ(β, h, x) and β c (x), it is evident that the location of the glass transition predicted by (7) coincides with β c (x = 0) and, thus, is completely determined by ϕ(β, h, 0) which agrees with a rescaled REM pressure [4]. In order to understand the qualitative behavior of the phase diagram and in particular the question of the stability of the glass phase in the QGREM with longitudinal field (4), it is thus convenient to restrict the discussion to the REM with constant fields, i.e., h = h and b = Γ for some positive constants h, Γ ≥ 0.…”

“…We turn to the interpolation argument for the left side in (7). Let U , U be two GREM processes with distribution functions A, A and pressures…”