Phase diagram and large deviations in the free-energy of mean-field spin-glasses

Abstract

We consider the probability distribution of large deviations in the spin-glass free energy for the Sherrington-Kirkpatrick mean field model, i.e. the exponentially small probability of finding a system with intensive free energy smaller than the most likely one. This result is obtained by computing $\Phi (n,T)=T\overline{Z^n}/n$, i.e. the average value of the partition function to the power $n$ as a function of $n$. We study in full details the phase diagram of $\Phi (n,T)$ in the $(n,T)$ plane computing in particular the stability of the replica-symmetric solution. At low temperatures we compute $\Phi (n,T)$ in series of $n$ and $\tau =T_c-T$ at high orders using the standard hierarchical ansatz and confirm earlier findings on the $O(n^5)$ scaling. We prove that the $O(n^5)$ scaling is valid at all orders and obtain an exact expression for the coefficient in term of the function $q(x)$. Resumming the series we obtain the large deviations probability at all temperatures. At zero temperature the analytical prediction displays a remarkable quantitative agreement with the numerical data. A similar computation for the simpler spherical model is also performed and the connection between large and small deviations is discussed.

Detailed study of the phase diagram and proof of the fifth order scaling at all temperatures, final version accepted on PRB

References in corpus (7)

• Large Deviations of Extreme Eigenvalues of Random Matrices
• Large Deviations in the Free-Energy of Mean-Field Spin-Glasses
• Can rare SAT formulas be easily recognized? On the efficiency of message passing algorithms for K-SAT at large clause-to-variable ratios
• Universality-class dependence of energy distributions in spin glasses
• Hysteretic optimization for the Sherrington-Kirkpatrick spin glass
• Double Criticality of the Sherrington-Kirkpatrick Model at T=0
• Free energy fluctuations and chaos in the Sherrington-Kirkpatrick model

Cited by in corpus (25)

• How many eigenvalues of a Gaussian random matrix are positive?
• Topology trivialization and large deviations for the minimum in the simplest random optimization
• Fluctuations of the free energy of the spherical Sherrington-Kirkpatrick model
• Replica Cluster Variational Method
• Index Distribution of Cauchy Random Matrices
• Simulations of Ground State Fluctuations in Mean-Field Ising Spin Glasses
• Large Deviations of the Free-Energy in Diluted Mean-Field Spin-Glass
• One-dimensional infinite component vector spin glass with long-range interactions
• Universality and Deviations in Disordered Systems
• Matching between typical fluctuations and large deviations in disordered systems : application to the statistics of the ground state energy in the SK spin-glass model
• Role of the Tracy-Widom distribution in the finite-size fluctuations of the critical temperature of the Sherrington-Kirkpatrick spin glass
• Ground State Properties of the Diluted Sherrington-Kirkpatrick Spin Glass
• Free energy fluctuations of the $2$-spin spherical SK model at critical temperature
• Fluctuations in the random-link matching problem
• A singular-potential random matrix model arising in mean-field glassy systems
• The Ising spin glass in finite dimensions: a perturbative study of the free energy
• Numerical Results for Spin Glass Ground States on Bethe Lattices: Gaussian Bonds
• Ground States of the Sherrington-Kirkpatrick Spin Glass with Levy Bonds
• Large deviations of the free energy in the p-spin glass spherical model
• Adversarial Satisfiability Problem
• Replica-symmetry breaking transitions in the large deviations of the ground-state of a spherical spin-glass
• Anomalous finite size corrections in random field models
• Sample-to-sample fluctuations and bond chaos in the $m$-component spin glass
• A replica trick for rare samples
• Order of Fluctuations of the Free Energy in the SK Model at Critical Temperature