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

Abstract

For the statistics of global observables in disordered systems, we discuss the matching between typical fluctuations and large deviations. We focus on the statistics of the ground state energy $E_0$ in two types of disordered models : (i) for the directed polymer of length $N$ in a two-dimensional medium, where many exact results exist (ii) for the Sherrington-Kirkpatrick spin-glass model of $N$ spins, where various possibilities have been proposed. Here we stress that, besides the behavior of the disorder-average $E_0^{av}(N)$ and of the standard deviation $\Delta E_0(N)\sim N^{{\omega }_f}$ that defines the fluctuation exponent ${\omega }_f$, it is very instructive to study the full probability distribution $\Pi (u)$ of the rescaled variable $u=\frac{E_0(N)-E_0^{av}(N)}{\Delta E_0(N)}$ : (a) numerically, the convergence towards $\Pi (u)$ is usually very rapid, so that data on rather small sizes but with high statistics allow to measure the two tails exponents ${\eta }_{\pm }$ defined as $\ln \Pi (u\to \pm \infty )\sim -|u{|}^{{\eta }_{\pm }}$. In the generic case $1<{\eta }_{\pm }<+\infty$, this leads to explicit non-trivial terms in the asymptotic behaviors of the moments $\overline{Z_N^n}$ of the partition function when the combination $[|n|N^{{\omega }_f}]$ becomes large (b) simple rare events arguments can usually be found to obtain explicit relations between ${\eta }_{\pm }$ and ${\omega }_f$. These rare events usually correspond to 'anomalous' large deviation properties of the generalized form $R(w_{\pm }=\frac{E_0(N)-E_0^{av}(N)}{N^{{\kappa }_{\pm }}})\sim e^{-N^{{\rho }_{\pm }}{\mathcal{R}}_{\pm }(w_{\pm })}$ (the 'usual' large deviations formalism corresponds to ${\kappa }_{\pm }=1={\rho }_{\pm }$).

10 pages, 4 figures

References in corpus (14)

• The large deviation approach to statistical mechanics
• Large Deviations of Extreme Eigenvalues of Random Matrices
• Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices
• Large Deviations of the Maximum Eigenvalue for Wishart and Gaussian Random Matrices
• Extreme value problems in Random Matrix Theory and other disordered systems
• Finite size corrections in the Sherrington-Kirkpatrick model
• Universality-class dependence of energy distributions in spin glasses
• Eigenvalue method to compute the largest relaxation time of disordered systems
• Probing the tails of the ground state energy distribution for the directed polymer in a random medium of dimension $d=1,2,3$ via a Monte-Carlo procedure in the disorder
• Probing tails of energy distributions using importance-sampling in the disorder with a guiding function
• Non equilibrium dynamics of disordered systems : understanding the broad continuum of relevant time scales via a strong-disorder RG in configuration space
• Exact Enumeration of Ground States in the Sherrington-Kirkpatrick Spin Glass
• Equilibrium of disordered systems : constructing the appropriate valleys in each sample via strong disorder renormalization in configuration space
• Statistics of first-passage times in disordered systems using backward master equations and their exact renormalization rules

Cited by in corpus (11)

• Critical Behaviour of the Number of Minima of a Random Landscape at the Glass Transition Point and the Tracy-Widom distribution
• How many eigenvalues of a Gaussian random matrix are positive?
• Typical versus averaged overlap distribution in Spin-Glasses : Evidence for the droplet scaling theory
• Random Transverse Field Ising Model in dimension $d>1$ : scaling analysis in the disordered phase from the Directed Polymer model
• Simulations of Ground State Fluctuations in Mean-Field Ising Spin Glasses
• Scaling of the largest dynamical barrier in the one-dimensional long-range Ising spin-glass
• One-dimensional Ising spin-glass with power-law interaction : real-space renormalization at zero temperature
• Chaos properties of the one-dimensional long-range Ising spin-glass
• A singular-potential random matrix model arising in mean-field glassy systems
• Adversarial Satisfiability Problem
• Replica-symmetry breaking transitions in the large deviations of the ground-state of a spherical spin-glass