Universality and Deviations in Disordered Systems

Abstract

We compute the probability of positive large deviations of the free energy per spin in mean-field Spin-Glass models. The probability vanishes in the thermodynamic limit as $P(\Delta f)\propto \exp [-N^2{L}_2(\Delta f)]$. For the Sherrington-Kirkpatrick model we find $L_2(\Delta f)=O(\Delta f{)}^{12/5}$ in good agreement with numerical data and with the assumption that typical small deviations of the free energy scale as $N^{1/6}$. For the spherical model we find $L_2(\Delta f)=O(\Delta f{)}^3$ in agreement with recent findings on the fluctuations of the largest eigenvalue of random Gaussian matrices. The computation is based on a loop expansion in replica space and the non-gaussian behaviour follows in both cases from the fact that the expansion is divergent at all orders. The factors of the leading order terms are obtained resumming appropriately the loop expansion and display universality, pointing to the existence of a single universal distribution describing the small deviations of any model in the full-Replica-Symmetry-Breaking class.

References in corpus (6)

• Large Deviations of Extreme Eigenvalues of Random Matrices
• Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices
• Large Deviations in the Free-Energy of Mean-Field Spin-Glasses
• Finite size corrections in the Sherrington-Kirkpatrick model
• Phase diagram and large deviations in the free-energy of mean-field spin-glasses
• Large Deviations of the Free-Energy in Diluted Mean-Field Spin-Glass

Cited by in corpus (10)

• Critical Behaviour of the Number of Minima of a Random Landscape at the Glass Transition Point and the Tracy-Widom distribution
• Interface free-energy exponent in the one-dimensional Ising spin glass with long-range interactions in both the droplet and broken replica symmetry regions
• Role of the Tracy-Widom distribution in the finite-size fluctuations of the critical temperature of the Sherrington-Kirkpatrick spin glass
• Fluctuations in the random-link matching problem
• A singular-potential random matrix model arising in mean-field glassy systems
• Classical Annealing of Sherrington-Kirkpatrick Spin Glass Using Suzuki-Kubo Mean-field Ising Dynamics
• Large deviations of the free energy in the p-spin glass spherical model
• Replica-symmetry breaking transitions in the large deviations of the ground-state of a spherical spin-glass
• Sample-to-sample fluctuations and bond chaos in the $m$-component spin glass
• A replica trick for rare samples