I will discuss the properties of the probability distribution P of a macro-variable [i.e. a quantity formed by the contributions of a large number of stochastic (micro-)variables], with emphasis on the phenomenon of "Condensation of fluctuations", the phenomenon whereby one of such micro-variables provides a macroscopic contribution to the global probability P.

The celebrated Fluctuation Theorem, quantifying the extent of breakage of time reversibility in nonequilibrium systems, pivots on the fact that all currents contributing to the dissipation rate of a system are known. What instead if only some currents are measurable? In general, the marginal probabilities of fewer currents do not satisfy the full fluctuation symmetry. However, still many claims can be sustained. We provide a general theory of fluctuation relations for the marginal p.d.f. of the currents of a Markov jump process, based on the mathematics of the "marginally time-reversed" generator. At the physical level, the theory has implications regarding the fluctuation-dissipation relations for systems that are only locally at equilibrium, i.e., such that certain currents vanish while all others are arbitrarily far from equilibrium. In particular, we are able to prove nonequilibrium Green-Kubo relations, the violation of the Onsager symmetry, and to provide higher-order signatures of nonequilibrium behavior.

The ubiquitous problem of cost function minimization greatly benefited in the last decades from the connection with ground state search in statistical mechanics systems. Physics' inspired Monte Carlo methods, such as Simulated Annealing and Parallel Tempering, and heuristics, such as Belief Propagation, proved to be highly effective in dealing with rough free energy landscapes. Here we present a slightly different mapping for the minimization problem, where the Gibbs weight is modified with the addition of a bias towards configurations having an high local entropy. For integer bias exponents, the resulting partition function is that of a replicated system with an attractive coupling among replicas. This leads to a general prescription that can be applied to many existing algorithms, resulting sometimes in huge improvements in their performances. I will discuss the performance of Simulated Annealing and Belief Propagation on the replicated system for the problems of neural networks and KSAT.
Bibliography: [1] C Baldassi, A Ingrosso, C Lucibello, L Saglietti and R Zecchina. Subdominant Dense Clusters Allow for Simple Learning and High Computational Performance in Neural Networks with Discrete Synapses. PRL 2015. [2] C Baldassi, A Ingrosso, C Lucibello, L Saglietti and R Zecchina. Local entropy as a measure for sampling solutions in Constraint Satisfaction Problems. JSTAT 2016.

Models of neural networks are a useful tool to investigate general mechanisms underlying dynamical regimes observed in neuronal systems, such as collective activity oscillations and the presence of critical avalanches in the cortex. We consider a reliable model for cortical neurons, a network of purely excitatory leaky integrate-and-fire neurons connected via the Tsodyks-Uziel-Markram (TUM) dynamical model for synaptic plasticity. We observe a transition to chaos as a function of the coupling strength and of the synaptic time scale, and we investigate this regime through a heterogeneous mean field approach. In the limit of a fully connected network, we are able to reduce the dynamics to a one-dimensional map that clarifies the mechanism underlying the transition. We observe that the onset of chaos in the fully connected model corresponds to a bursty phase on a disordered topology. We characterize the latter phase as strongly correlated and potentially carrying a larger amount of information than quasi-synchronous and asynchronous regimes.

Training of neural networks in which the synaptic connections strengths are discretized and their precision is limited to only a few bits has long been considered a challenging task even for the simplest neural architectures: local search algorithms tend to be easily trapped in local minima, and equilibrium statistical analysis shows that global optima are typically isolated ("golf-course scenario"). However, biological experiments show that the precision of brain's synapses does not exceed very few bits (and may even be as low as 1 bit). Furthermore, machine learning applications would greatly benefit from reduced requirements. We performed a large deviations analysis which shows that there exist peculiar dense regions in the space of synaptic states which account for the possibility of learning under these constraints. These regions are characterized by a large local entropy, such that: 1) they are accessible to very simple and efficient heuristic algorithms which exploit their characteristics; 2) the optima are very wide and thus robust; 3) they have good generalization properties. The analytical results give us an "effective capacity" measure which saturates fast with the number of synaptic values and thus indicates that very few bits are indeed sufficient for effective learning. Our numerical observations match the theoretical results where available, and indicate that the scenario extends to complex multi-layer neural architectures trained on real-world data (potentially providing a framework to explain the success of deep learning techniques). The analysis may also be extended to other models.
Bibliography: [1] C Baldassi, A Ingrosso, C Lucibello, L Saglietti and R Zecchina. Subdominant Dense Clusters Allow for Simple Learning and High Computational Performance in Neural Networks with Discrete Synapses. PRL 2015. [2] C Baldassi, F Gerace, C Lucibello, L Saglietti and R Zecchina. Learning may need only few bits of synaptic precision. http://arxiv.org/abs/1602.04129

We study the force-velocity relation of a tracer particle driven by an external force in two different systems. First, we present analytical results in the case of a lattice gas model, where the tracer moves in an environment of mobile hard-core obstacles. We discuss in particular the surprising effect of negative differential mobility (NDM), namely a nonmonotonic behavior of the tracer velocity as a function of the external force. Then, we discuss the case of an inertial particle advected by a laminar flow, where the nonlinear response of the tracer shows a very rich phenomenology. An explanation of the observed behaviors in terms of a generalized Einstein relation is proposed.

The fractional quantum Hall effect still poses challenges to contemporary physics. Recent experiments point toward a fractal scenario for the Hall resistivity as a function of the magnetic field. We show that, in the thin-torus limit, the Hamiltonian for interacting electrons in a strong magnetic field, restricted to the lowest Landau level, can be mapped onto a 1D lattice gas with repulsive interactions, with the magnetic field playing the role of the chemical potential. The statistical mechanics of such models leads to interpret the sequence of Hall plateaux as a fractal phase diagram, whose landscape shows a qualitative agreement with experiments.

We study current-current correlation in an electronic analog of a beam splitter realized with edge channels of a fractional quantum Hall liquid at Laughlin filling fractions. In analogy with the known result for chiral electrons , if the currents are measured at points located after the beam splitter, we find that the zero frequency equilibrium correlation vanishes due to the chiral propagation along the edge channels. Furthermore, we show that the current-current correlation,normalized to the tunneling current, exhibits clear signatures of the Laughlin quasi-particles fractional statistics.

We use citation data of scientific articles produced by individual nations in different scientific domains to build a bipartite country - scientific domains network to determine the structure and efficiency of national research systems [1]. We characterize the scientific fitness of each nation - that is, the competitiveness of its research system - and the complexity of each scientific domain by means of a non-linear iterative algorithm [2] able to assess quantitatively the advantage of scientific diversification. We find that technological leading nations, beyond having the largest production of scientific papers and the largest number of citations, do not specialize in a few scientific domains. Rather, they diversify as much as possible their research system. On the other side, less developed nations are competitive only in scientific domains where also many other nations are present. Diversification thus represents the key element that correlates with scientific and technological competitiveness. A remarkable implication of this structure of the scientific competition is that the scientific domains playing the role of markers of national scientific competitiveness are those not necessarily of high technological requirements, but rather addressing the most sophisticated needs of the society. We complement this analysis with a correlation study between the scientific impact of a nation with a normalized measure of RD funds and the level of internationalization [3].
[1] G. Cimini, A. Gabrielli, F. Sylos Labini (2014), PLoS ONE 9(12), e113470. [2] A. Tacchella et al. (2013), Sci. Rep. 2, 723. [3] G. Cimini, A. Zaccaria, A. Gabrielli (2016), J. of Informetrics 10, 200.

We propose a similarity measure between portfolios with possibly very different numbers of assets and apply it to a historical database of institutional holdings ranging from 1999 to the end of 2013. The resulting portfolio similarity measure increased steadily before the global financial crisis, and reached a maximum when the crisis occurred. We argue that the nature of this measure implies that liquidation risk from fire sales was maximal at that time. After a sharp drop in 2008, portfolio similarity resumed its growth in 2009, with a notable acceleration in 2013, reaching levels not seen since 2007.