Let {E<sub>σ</sub> (N)}<sub>σ∈ΣN</sub> be a family of |Σ<sub>N</sub>| = 2<sup>N</sup> centered unit Gaussian random variables defined by the covariance matrix C<sub>N</sub> of elements c<sub>N</sub>(σ, τ): = Av(E<sub>σ</sub>(N)E<sub>τ</sub>(N)) and H<sub>N</sub>(σ) = -√NE<sub>σ</sub>(N) the corresponding random Hamiltonian. Then the quenched thermodynamical limit exists if, for every decomposition N = N<sub>1</sub> + N<sub>2</sub>, and all pairs (σ, τ) ∈ Σ<sub>N</sub> × Σ<sub>N</sub>: c<sub>N</sub>(σ, τ) ≤ N<sub>1</sub>/N c<sub>N1</sub>(π<sub>1</sub>(τ), π<sub>1</sub>(τ)) + N<sub>2</sub>/N c<sub>N2</sub>(π<sub>2</sub>(σ), π<sub>2</sub>(τ)), where π<sub>k</sub> (τ), k = 1, 2 are the projections of σ ∈ Σ<sub>N</sub> into Σ<sub>Nk</sub>. The condition is explicitly verified for the Sherrington-Kirkpatrick, the even p-spin, the Derrida REM and the Derrida-Gardner GREM models.
Thermodynamical limit for correlated Gaussian random energy models / P., Contucci; M., Degli Esposti; Giardina', Cristian; S., Graffi. - In: COMMUNICATIONS IN MATHEMATICAL PHYSICS. - ISSN 0010-3616. - STAMPA. - 236:1(2003), pp. 55-63. [10.1007/s00220-003-0803-y]
Thermodynamical limit for correlated Gaussian random energy models
GIARDINA', Cristian;
2003
Abstract
Let {Eσ (N)}σ∈ΣN be a family of |ΣN| = 2N centered unit Gaussian random variables defined by the covariance matrix CN of elements cN(σ, τ): = Av(Eσ(N)Eτ(N)) and HN(σ) = -√NEσ(N) the corresponding random Hamiltonian. Then the quenched thermodynamical limit exists if, for every decomposition N = N1 + N2, and all pairs (σ, τ) ∈ ΣN × ΣN: cN(σ, τ) ≤ N1/N cN1(π1(τ), π1(τ)) + N2/N cN2(π2(σ), π2(τ)), where πk (τ), k = 1, 2 are the projections of σ ∈ ΣN into ΣNk. The condition is explicitly verified for the Sherrington-Kirkpatrick, the even p-spin, the Derrida REM and the Derrida-Gardner GREM models.Pubblicazioni consigliate
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