Thermodynamical Limit for Correlated Gaussian Random Energy Models

Abstract

Let $\{E_{\s}(N)\}_{\s\inΣ_N}$ be a family of $|{\Sigma }_N|=2^N$ centered unit Gaussian random variables defined by the covariance matrix $C_N$ of elements $\displaystyle c_N(\s,τ):=\av{E_{\s}(N)E_τ(N)}$, and $H_N(\s) = - \sqrt{N} E_{\s}(N)$ the corresponding random Hamiltonian. Then the quenched thermodynamical limit exists if, for every decomposition $N=N_1+N_2$, and all pairs $(\s,\t)\in Σ_N\times Σ_N$: $$ c_N(\s,τ)\leq \frac{N_1}{N} c_{N_1}(π_1(\s),π_1(τ))+ \frac{N_2}{N} c_{N_2}(π_2(\s),π_2(τ)) $$ where $π_k(\s), k=1,2$ are the projections of $\s\inΣ_N$ into ${\Sigma }_{N_k}$. The condition is explicitly verified for the Sherrington-Kirckpatrick, the even $p$-spin, the Derrida REM and the Derrida-Gardner GREM models.

15 pages, few remarks and two references added. To appear in Commun. Math. Phys

Cited by in corpus (19)

• An Extended Variational Principle for the SK Spin-Glass Model
• Spin-Glass Stochastic Stability: a Rigorous Proof
• The infinite volume limit in generalized mean field disordered models
• Finite-range spin glasses in the Kac limit: free energy and local observables
• Griffiths inequalities for the Gaussian spin glass
• Replica equivalence in the Edwards-Anderson model
• On the Surface Pressure for the Edwards-Anderson Model
• Cluster Percolation in the Two-Dimensional Ising Spin Glass
• Variational Bounds for the Generalized Random Energy Model
• Thermodynamic Limit for Finite Dimensional Classical and Quantum Disordered Systems
• Fluctuations of the partition function in the GREM with external field
• Contributions to Random Energy Models
• Thermodynamic Limit for Mean-Field Spin Models
• Dilution Robustness for Mean Field Ferromagnets
• Convex Replica Simmetry Breaking From Positivity and Thermodynamic Limit
• Rigorous Results for Hierarchical Models of Structural Glasses
• Energy Landscape Statistics of the Random Orthogonal Model
• Remarks on the interpolation method
• Finite-size corrections for the attractive mean-field monomer-dimer model