Stochastic Stability: a Review and Some Perspectives

A review of the stochastic stability property for the Gaussian spin glass models is presented and some perspectives discussed.Comment: 12 pages, typos corrected, references added. To appear in Journal of Statistical Physics, Special Issue for the 100th Statistical Mechanics Meetin

Inequalities for the Local Energy of Random Ising Models

We derive a rigorous lower bound on the average local energy for the Ising model with quenched randomness. The result is that the lower bound is given by the average local energy calculated in the absence of all interactions other than the one under consideration. The only condition for this statement to hold is that the distribution function of the random interaction under consideration is symmetric. All other interactions can be arbitrarily distributed including non-random cases. A non-trivial fact is that any introduction of other interactions to the isolated case always leads to an increase of the average local energy, which is opposite to ferromagnetic systems where the Griffiths inequality holds. Another inequality is proved for asymmetrically distributed interactions. The probability for the thermal average of the local energy to be lower than that for the isolated case takes a maximum value on the Nishimori line as a function of the temperature. In this sense the system is most stable on the Nishimori line.Comment: 10 pages. Submitted to J. Phys. Soc. Jp

On the Stability of the Quenched State in Mean Field Spin Glass Models

While the Gibbs states of spin glass models have been noted to have an erratic dependence on temperature, one may expect the mean over the disorder to produce a continuously varying ``quenched state''. The assumption of such continuity in temperature implies that in the infinite volume limit the state is stable under a class of deformations of the Gibbs measure. The condition is satisfied by the Parisi Ansatz, along with an even broader stationarity property. The stability conditions have equivalent expressions as marginal additivity of the quenched free energy. Implications of the continuity assumption include constraints on the overlap distribution, which are expressed as the vanishing of the expectation value for an infinite collection of multi-overlap polynomials. The polynomials can be computed with the aid of a "real"-replica calculation in which the number of replicas is taken to zero.Comment: 17 pages, LaTex, Revised June 5, 199

Interaction Flip Identities for non Centered Spin Glasses

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

Convex Replica Simmetry Breaking From Positivity and Thermodynamic Limit

Consider a correlated Gaussian random energy model built by successively adding one particle (spin) into the system and imposing the positivity of the associated covariance matrix. We show that the validity of a recently isolated condition ensuring the existence of the thermodynamic limit forces the covariance matrix to exhibit the Parisi replica symmetry breaking scheme with a convexity condition on the matrix elements.Comment: 11 page

Thermodynamic Limit for Spin Glasses. Beyond the Annealed Bound

Using a correlation inequality of Contucci and Lebowitz for spin glasses, we demonstrate existence of the thermodynamic limit for short-ranged spin glasses, under weaker hypotheses than previously available, namely without the assumption of the annealed bound.Comment: 8 page

Scaling Limits for Multispecies Statistical Mechanics Mean-Field Models

We study the limiting thermodynamic behavior of the normalized sums of spins in multi-species Curie-Weiss models. We find sufficient conditions for the limiting random vector to be Gaussian (or to have an exponential distribution of higher order) and compute the covariance matrix in terms of model parameters.Comment: 21 page

Monotonicity and Thermodynamic Limit for Short Range Disordered Models

If the variance of a short range Gaussian random potential grows like the volume its quenched thermodynamic limit is reached monotonically.Comment: 2 references adde

Spin-Glass Stochastic Stability: a Rigorous Proof

We prove the property of stochastic stability previously introduced as a consequence of the (unproved) continuity hypothesis in the temperature of the spin-glass quenched state. We show that stochastic stability holds in beta-average for both the Sherrington-Kirkpatrick model in terms of the square of the overlap function and for the Edwards-Anderson model in terms of the bond overlap. We show that the volume rate at which the property is reached in the thermodynamic limit is V^{-1}. As a byproduct we show that the stochastic stability identities coincide with those obtained with a different method by Ghirlanda and Guerra when applyed to the thermal fluctuations only.Comment: 12 pages, revised versio

Stochastic Stability and the Spin Glass Phase. The State of the Art for Mean Field and Finite Dimensional Models

Some invariances under perturbations of the spin glass phase are introduced, their proofs outlined and their consequences illustrated as factorisation rules for the overlap distribution. A comparison between the state of the art for mean field and finite dimensional models is shortly discussed.Comment: Invited address at the International Congress on Mathematical Physics, Aalborg 2012, Denmar