Elenco seminari del ciclo di seminari
“MATHEMATICAL PHYSICS OF DISORDERED SYSTEMS”

We give a short review about the replica trick in the frame of a simple interpolation scheme on the number of replicas. In this way we avoid any problem with analytic continuation and recognise that replica symmetry breaking is in fact characterised by a phase transition. We give application of the method to the case of the Random Energy Model, and the Sherrington-Kirkpatrick mean field spin glass model.

Spin glasses are disordered spin systems initially invented by theoretical physicists with the aim of understanding some strange magnetic properties of certain alloys. In particular, over the past decades, the study of the Sherrington-Kirkpatrick (SK) mean-field model via the replica method has received great attention. In this talk, I will discuss another approach to studying the SK model proposed by Thouless-Anderson-Palmer (TAP). I will explain how the generalized TAP correction appears naturally and give the corresponding generalized TAP representation for the free energy. Based on a joint work with D. Panchenko and E. Subag.

The ferromagnetic Ising model is a paradigmatic model of statistical physics used to study phase transitions in lattice systems. In this talk I shall consider the setting where the regular spatial structure is replaced by a random graph, which is often used to model complex networks. I shall treat both the case where the graph is essentially frozen (quenched setting) and the case where instead it is rapidly changing (annealed setting). I shall prove that quenched and annealed may have different critical temperatures, provided the graph has sufficient inhomogeneity. I shall also discuss how universal results (law of large numbers, central limit theorems, critical exponents) are affected by the disorder in the spatial structure. The picture that I will present emerges from several joint works, involving V.H. Can, S. Dommers, C. Giberti, R.van der Hofstad and M.L.Prioriello.

Mean-field methods fail to reconstruct the parameters of the model when the dataset is clusterized. This situation is found at low temperatures because of the emergence of multiple thermodynamic states. The paradigmatic Hopfield model is considered in a teacher student scenario as a problem of unsupervised learning with Restricted Boltzmann Machines (RBM). For different choices of the priors on units and weights, the replica symmetric phase diagram of random RBM’s is analyzed and in particular the paramagnetic phase boundary is presented as directly related to the optimal size of the training set necessary for a good generalization. The connection between the direct and inverse problem is pointed out by showing that inference can be efficiently performed by suitably adapting both standard learning technique and standard approaches to the direct problem.

I will present a framework for rigorously establishing the information-theoretic limits in high-dimensional generalized linear models (GLMs). The GLM includes as special cases plethora of important models in signal processing (compressed-sensing, phase retrieval etc), communications, but also in learning such as the famous perceptron neural network. Many instances of GLMs have been analyzed in the statistical physics literature, in particular thanks to the heuristic replica method developed in the context of spin glasses. I will discuss a recent technique called « adaptive interpolation method » that allows to vindicate the statistical approach in a unified manner, as well as recent findings about the rich algorithmic behaviors encountered in such models.

We will discuss why the extremal process of oriented FPP on the hypercube converges to a Cox process with exponential intensity (arXiv:1804.03117 and 1808.04598)

We consider a mean field spin system with random interaction in a multiscale thermal equilbrium, namely when the thermodynamic equilibrium is reached through successive hierarchical thermalization. This model apperead in physical literature un the 90's as a possible framework describing thermodynamic properties of systems out of standard Gibbs equilibrium. We show how to construct a variational problem that gives a representation of the thermodynamic limit of the pressure density (generating functional). We present the main ideas behind the proof and discuss possible applications of the result.