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

Modeling Society with Statistical Mechanics: an Application to Cultural Contact and Immigration

We introduce a general modeling framework to predict the outcomes, at the population level, of individual psychology and behavior. The framework prescribes that researchers build a cost function that embodies knowledge of what trait values (opinions, behaviors, etc.) are favored by individual interactions under given social conditions. Predictions at the population level are then drawn using methods from statistical mechanics, a branch of theoretical physics born to link the microscopic and macroscopic behavior of physical systems. We demonstrate our approach building a model of cultural contact between two cultures (e.g., immigration), showing that it is possible to make predictions about how contact changes the two cultures

Local Order at Arbitrary Distances in Finite-Dimensional Spin-Glass Models

For a finite dimensional spin-glass model we prove local order at low temperatures for both local observables and for products of observables at arbitrary mutual distance. When the Hamiltonian includes the Edwards-Anderson interaction we prove "bond" local order, when it includes the random-field interaction we prove "site" local order

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

Bipartite Mean Field Spin Systems. Existence and Solution

A mean field spin system consisting two interacting groups each with homogeneous interaction coefficients is introduced and studied. Existence of the thermodynamic limit is shown by an asymptotic sub-addittivity method and factorization of correlation functions is proved almost everywhere. The free energy solution of the model is obtained by upper and lower bounds and by showing that their difference vanishes for large volumes.Comment: 1 Figur

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

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

Correlation Inequalities for Spin Glasses

We prove a correlation type inequality for spin systems with quenched symmetric random interactions. This gives monotonicity of the pressure with respect to the strength of the interaction for a class of spin glass models. Consequences include existence of the thermodynamic limit for the pressure and bounds on the surface pressure. We also describe other conjectured inequalities for such systems

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

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