# Research Activity

### Large Deviation approach to Localization and Ergodicity Breaking

One of the historically most challenging problems in statistical mechanics is that of ergodicity breaking. Besides systems with quenched disorder as spin glasses, this phenomenon can be found in both classical and quantum systems with many degrees of freedom in certain regimes. In particular in quantum systems ergodicity breaking appears in the form of localization, that is, the wavefunction is localized in space rather than filling it uniformly. We have recently developed a probabilistic approach based on the theory of Large Deviations to characterize the transition from the delocalized to the localized phase in a certain class of quantum system, i.e. Bose-Einstein condensates. This approach is quite promising: an interesting starting project is about extending this technique to the study of ergodicity breaking in the one-dimensional Fermi-Pasta-Ulam model, a classical non-integrable Hamiltonian system of particles coupled by non-linear springs.

### Statistical mechanics of disordered systems

One among the most striking advances in the statistical mechanics of the last 40 years is the theory of ergodicity breaking in systems with quenched disorder as spin glasses. This theory, which goes under the name of “replica symmetry breaking”, is exact in the limit of large system size only for a certain class of models, known as mean-field models. In particular, from the point of view of complex networks, mean-field models are those for which the web of interactions is represented by a complete graph. Only for such models the theory of “replicas” works exactly. Since more realistic models are characterized by a diluted network of interactions our effort is to develop methods to analytically solve the thermodynamics of disordered systems when the complete graph structure is diluted. This line of research is focused, on the one hand, on the extension of the theory of replicas to system on diluted networks and, on the other hand, to the extension of the so-called “cavity” method introduced by Mezard and Parisi, initially conceived for sparse networks, to dense networks. The “cavity” method and the related Belief Propagation equations provide someone with a much more transparent probabilistic interpretation of the breaking of ergodicity than the replica theory.

The methods we are developing, beside their intrinsic interest for the investigation of disordered systems, are for instance valuable tools for Inference and Learning problems or can be applied to the study of realistic system as light confined in disordered materials (random lasers).

Research in this field is both analytical and numerical: numerical skills are highly appreciated together with the attitude to analytical solutions.

### Integrable systems and the Foundations of Statistical Mechanics

We are experiencing nowdays a renowned interests in fundamental problems in statistical mechanics, in particular how thermal equilibrium is approached in integrable systems, both classical and quantum. Which is the key ingredient which allows one to use a statistical mechanics probabilistic description for a physical system? According to Kinchin this is just a matter of having a large number of degrees of freedom and to do the appropriate choice of variables. A paradigmatic example is that of the Fermi-Pasta-Ulam model, i.e. a chain of particles connected by weakly non-linear springs: starting the Hamiltonian dynamics from a very atypical initial condition one finds a fast equilibration for the particle degrees of freedom whereas Fourier modes equilibrate on a much longer time-scale. This is the typical example of a situation where the degree of ergodicity of the system depends upon which degrees of freedom are considered, in this case either the particles coordinates or the amplitudes of Fourier modes. Beside the choice of variables, the study of the Fermi-Pasta-Ulam model poses then another central problem: how much the lack of thermal equilibrium is due to the proximity in parameter space to integrable models? The aim at answering to these and other question is the heart of our research project. This line of research is clearly highly conceptual but is for interest for many hot areas in quantum and classical mechanics: Eigenstate Thermalization Hypothesis, Integrability in classical and quantum systems, Ergodicity breaking in quantum systems, Many-Body Localization, Entanglement Entropy, Inequivalence of Statistical Ensembles.