Abstract:
The determination of the conductivity of a deterministic or stochastic classical system coupled to reservoirs at its ends can in general be mapped onto the problem of computing the stiffness (the `energy' cost of twisting the boundaries) of a quantum-like operator. The nature of the coupling to the reservoirs determines the details of the mechanical coupling of the torque at the ends.

Abstract:
Starting from the study of a linear combination of multi-overlaps which can be rigorously shown to vanish for large systems we numerically analyze the factorization properties of the link-overlaps multi-distribution for the 3D Gaussian Edward-Anderson spin-glass model. We find evidence of a pure factorization law for the multi-correlation functions. For instance the quantity [ - ]/ tends to zero at increasing volumes. We also perform the same analysis for the standard overlap for which instead the lack of factorization persists increasing the size of the system. The necessity of a better understanding of the mutual relation between the two overlaps is pointed out.

Abstract:
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.

Abstract:
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.

Abstract:
Non-equilibrium steady states are subject to intense investigations but still poorly understood. For instance, the derivation of Fourier law in Hamiltonian systems is a problem that still poses several obstacles. In order to investigate non-equilibrium systems, stochastic models of energy-exchange have been introduced and they have been used to identify universal properties of non-equilibrium. In these notes, after a brief review of the problem of anomalous transport in 1-dimensional Hamiltonian systems, some boundary-driven interacting random systems are considered and the "duality approach" to their rigorous mathematical treatment is reviewed. Duality theory, of which a brief introduction is given, is a powerful technique to deal with Markov processes and interacting particle systems. The content of these notes is mainly based on the papers [10, 11, 12].

Abstract:
For a general spin glass model with asymmetric couplings we prove a family of identities involving expectations of generalized overlaps and magnetizations in the quenched state. Those identities holds pointwise in the Nishimori line and are reached at the rate of the inverse volume while, in the general case, they can be proved in integral average.

Abstract:
We study the properties of fluctuation for the free energies and internal energies of two spin glass systems that differ for having some set of interactions flipped. We show that their difference has a variance that grows like the volume of the flipped region. Using a new interpolation method, which extends to the entire circle the standard interpolation technique, we show by integration by parts that the bound imply new overlap identities for the equilibrium state. As a side result the case of the non-interacting random field is analyzed and the triviality of its overlap distribution proved.

Abstract:
We consider spin glass models with non-centered interactions and investigate the effect, on the random free energies, of flipping the interaction in a subregion of the entire volume. A fluctuation bound obtained by martingale methods produces, with the help of integration by parts technique, a family of polynomial identities involving overlaps and magnetizations.

Abstract:
We introduce a numerical procedure to evaluate directly the probabilities of large deviations of physical quantities, such as current or density, that are local in time. The large-deviation functions are given in terms of the typical properties of a modified dynamics, and since they no longer involve rare events, can be evaluated efficiently and over a wider ranges of values. We illustrate the method with the current fluctuations of the Totally Asymmetric Exclusion Process and with the work distribution of a driven Lorentz gas.

Abstract:
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.