Ricerc@Sapienza

In recent years Italy has been involved in massive migration flows and, con- sequently, migrant integration is becoming a urgent political, economic and social issue. In this paper we apply quantitative methods, based on probability theory and statistical mechanics, to study the relative integration of migrants in Italy.

In recent decades, statistical mechanics of disordered systems (mainly spin-glasses) has become one of the main tools to investigate complex systems, probably due to the celebrated Replica Symmetry Breaking scheme of Parisi Theory and its deep implications. In this work we consider the analog bipartite spin-glass (or real-valued restricted Boltzmann machine in a neural-network jargon), whose variables (those quenched as well as those dynamical) share standard Gaussian distributions.

The standard Hopfield model for associative neural networks accounts for biological Hebbian learning and acts as the harmonic oscillator for pattern recognition, however its maximal storage capacity is α∼0.14, far from the theoretical bound for symmetric networks, i.e. α=1.

In this work, we introduce and investigate the properties of the “relativistic” Hopfield model endowed with temporally correlated patterns.
First, we review the “relativistic” Hopfield model and we briefly describe the experimental evidence underlying correlation among patterns.
Then, we face the study of the resulting model exploiting statistical-mechanics tools in a low-load regime. More precisely, we prove the
existence of the thermodynamic limit of the related free energy and we derive the self-consistence equations for its order parameters. These

In this paper we adapt the broken replica interpolation technique (developed by Francesco Guerra to deal with the Sherrington-Kirkpatrick model, namely a pairwise mean-field spin-glass whose couplings are i.i.d. standard Gaussian variables) in order to work also with the Hopfield model (i.e. a pairwise mean-field neural-network whose couplings are drawn according to Hebb's learning rule): This is accomplished by grafting Guerra's telescopic averages on the transport equation technique, recently developed by some of the authors.

We consider a three-layer Sejnowski machine and show that features learnt via contrastive divergence have a dual representation as patterns in a dense associative memory of order P = 4. The latter is known to be able to Hebbian store an amount of patterns scaling as NP -1, where N denotes the number of constituting binary neurons interacting P wisely.