Variational Bounds for the Generalized Random Energy Model

We compute the pressure of the random energy model (REM) and generalized random energy model(GREM) by establishing variational upper and lower bounds. For the upper bound, we generalize Guerra's ``broken replica symmetry bounds",and identify the random probability cascade as the appropriate random overlap structure for the model. For the REM the lower bound is obtained, in the high temperature regime using Talagrand's concentration of measure inequality, and in the low temperature regime using convexity and the high temperature formula. The lower bound for the GREM follows from the lower bound for the REM by induction. While the argument for the lower bound is fairly standard, our proof of the upper bound is new.Comment: 24 page

The Ghirlanda-Guerra Identities

If the variance of a Gaussian spin-glass Hamiltonian grows like the volume the model fulfills the Ghirlanda-Guerra identities in terms of the normalized Hamiltonian covariance.Comment: 18 page

Stability of the Spin Glass Phase under Perturbations

We introduce and prove a novel linear response stability theory for spin glasses. The new stability under suitable perturbation of the equilibrium state implies the whole set of structural identities that characterize the spin glass phase.Comment: 5 pages. Changed abstract, corrected typos, added reference

The Non-Equilibrium Ising Model in Two Dimensions: a Numerical Study

In this paper, we study the boundary-driven ferromagnetic Ising model in two dimensions. In this non-equilibrium setting, in the low temperature region, the Ising model has phase separation in the presence of a current. We investigate, by means of numerical simulations, Kawasaki dynamics with magnetization reservoirs. The results show that, in the stationary non-equilibrium state, the Ising model may have uphill diffusion and magnetization profiles with three discontinuities. These results complement the results of a previous paper by Colangeli, Giberti, Vernia and the present author [9]. They also allow to state a full picture of the hydrodynamic limit

Relative entropy and waiting times for continuous-time Markov processes

For discrete-time stochastic processes, there is a close connection between return/waiting times and entropy. Such a connection cannot be straightforwardly extended to the continuous-time setting. Contrarily to the discrete-time case one does need a reference measure and so the natural object is relative entropy rather than entropy. In this paper we elaborate on this in the case of continuous-time Markov processes with finite state space. A reference measure of special interest is the one associated to the time-reversed process. In that case relative entropy is interpreted as the entropy production rate. The main results of this paper are: almost-sure convergence to relative entropy of suitable waiting-times and their fluctuation properties (central limit theorem and large deviation principle).Comment: 17 page

Finding Minima in Complex Landscapes: Annealed, Greedy and Reluctant Algorithms

We consider optimization problems for complex systems in which the cost function has a multivalleyed landscape. We introduce a new class of dynamical algorithms which, using a suitable annealing procedure coupled with a balanced greedy-reluctant strategy drive the systems towards the deepest minimum of the cost function. Results are presented for the Sherrington-Kirkpatrick model of spin-glasses.Comment: 30 pages, 12 figure

Dualities in population genetics: a fresh look with new dualities

We apply our general method of duality, introduced in [Giardina', Kurchan, Redig, J. Math. Phys. 48, 033301 (2007)], to models of population dynamics. The classical dualities between forward and ancestral processes can be viewed as a change of representation in the classical creation and annihilation operators, both for diffusions dual to coalescents of Kingman's type, as well as for models with finite population size. Next, using SU(1,1) raising and lowering operators, we find new dualities between the Wright-Fisher diffusion with $d$ types and the Moran model, both in presence and absence of mutations. These new dualities relates two forward evolutions. From our general scheme we also identify self-duality of the Moran model.Comment: 36 pages, to appear on Stochastic Processes and their Application

Overlap Equivalence in the Edwards-Anderson Model

We study the relative fluctuations of the link overlap and the square standard overlap in the three dimensional Gaussian Edwards-Anderson model with zero external field. We first analyze the correlation coefficient and find that the two quantities are uncorrelated above the critical temperature. Below the critical temperature we find that the link overlap has vanishing fluctuations for fixed values of the square standard overlap and large volumes. Our data show that the conditional variance scales to zero in the thermodynamic limit. This implies that, if one of the two random variables tends to a trivial one (i.e. delta-like distributed), then also the other does and, by consequence, the TNT picture should be dismissed. We identify the functional relation among the two variables using the method of the least squares which turns out to be a monotonically increasing function. Our results show that the two overlaps are completely equivalent in the description of the low temperature phase of the Edwards-Anderson model.Comment: Latex file, 8 Pages, 4 Figures. To appear in: Physical Review Letter