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“WORKSHOP "MATHEMATICAL PHYSICS AND BEYOND"”

By relying upon tools of statistical mechanics of spin glasses, in this talk I will focus on Hebbian neural networks interacting in an heteroassociative manner to show that the overall network as a whole shows computationally capabilities that are lost within a single neural network. In particular I will show how these networks naturally disentangle spurious states recovering the original patterns forming these mixtures, thus providing a novel way of performing challenging pattern recognition tasks. The theory will be developed in the standard random setting then applications will be performed on structured datasets as the harmonic melodies.

The method of Replica Symmetry Breaking is considered in the frame of replica interpolation, where it leads to a kind of phase transition. Applications are given for the Random Energy Model and for The Sherrington Kirkpatrick model. The results show some unexpected surprises.

In this talk I will recall some work on the application of statistical physics techniques to social data and discuss some recent perspectives on data from historical archives.

I propose a personal overview of my collaboration with Pierluigi since our first meeting in 2003. I’ll review our numerical work on the glassy phase of finite dimensional spin glasses; in particular, overlap equivalence, ultrametricity, clustering property of overlap and monotonicity of the correlation functions will be considered. I’ll also present our research on the inverse problem in some mean field models with applications to the social sciences.

An assumption that typically pervades the study of spin glasses is that of independent and identically distributed random variables. In the celebrated Sherrington-Kirkpatrick model this is manifest in the distribution of the quenched disorder. This homogeneity creates a system whose particles are indistinguishable from one another, namely they can be arbitrarily permuted without changing the thermodynamical features of the model. Breaking identicality in the quenched disorder also breaks this global permutation symmetry, with the possibility of leaving it intact only in smaller subgroups of particles involved. The latter procedure leads to the definition of multispecies spin glasses, which are typically harder to analyse. In my talk I will give an overview of the cases we can solve, with a particular focus on multispecies models on the Nishimori Line, that is a particular region of their phase space where they have a clear correspondence with high dimensional inference problems, and concentration of the order parameters holds despite the presence of quenched disorder.

The ferromagnetic Ising spin model is often used to model second-order phase transitions and the continuous emergence of order. We consider this model on a random graph, where the additional randomness provided by the graph gives a rich picture with a host of surprises. We identify similarities and differences between the quenched and annealed Ising model. We find that the annealed critical temperature is highly model-dependent, even in the case of graphs that are asymptotically equivalent (such as different versions of the simple Erdös-Rényi random graph). The quenched critical temperature is instead the same for all locally tree-like graphs. Moreover, in the presence of inhomogeneities that produce a fat-tail degree distribution, the difference between quenched and annealed becomes even more substantial, leading in some cases to different universality classes and different critical exponents. The annealed properties depend sensitively on whether the total number of edges of the underlying random graph is fixed, or is allowed to fluctuate. If time allows preliminary results on the annealed Potts model, displaying a first-order phase transition, will also be discussed. [This talk is based on several joint works with Hao Can, Sander Dommers, Claudio Giberti, Remco van der Hofstad and Maria Luisa Prioriello. The preliminary work on Potts models also involves Neeladri Maitra and benefited from discussions with Guido Janssen.]

We review some recent advances in the rigorous analysis of mean field spin glasses. In particular we show how Parisi's theory can be generalized in the case where the spin-spin interaction is not described by i.i.d. random variables but, to some extent, it's of mean field type. We will focus on the multipecies SK model and the multiscale SK model, presenting the variational formulas for the free energy with a sketch of the proofs.

Under the same assumptions and level of rigour as the previous work of Franz, Mezard, Parisi and Peliti, one can show that the ultrametric solution for the equilibrium measure holds if and only if the system's dynamics spontaneously split into widely separated timescales with only one temperature per timescale for all observables.

We discuss a family of multispecies ferromagnetic Ising models on multiregular random graphs. In the large volume limit, thermodynamic quantities are related to the solution of a belief propagation (BP) fixed point equation. A phase transition is identified and the critical region is determined by the spectral radius of a finite-dimensional matrix.

Chaotic behavior and Stochastic Stability are two faces of the same RSB coin. In this talk I will discuss the universal properties of chaos against a small magnetic field in spin glasses. The introduction of a small field in a spin-glass modifies the weights of the equilibrium states. Using the fact that the magnetizations form a Gaussian process on the UM tree we can study the progressive decorrelation of the system in the field from the system without the field. We can then provide predictions on chaos that only depend on the Parisi function $P(q)$ in absence of the field. I will discuss in detail the simple case of the 1RSB, where extreme value statistics allow to completely solve the problem. In the full RSB case it is possible in principle to solve the problem through Parisi-like PDE, however, we found it more practical to simulate the infinite-system stochastic process implied by RSB theory. Getting a function $P(q)$ as input, we can generate weighted random trees using the Bolthausen-Snitman coalescent, reweight the states according to the values of their magnetization. We compare the theoretical predictions with direct simulations of Bethe-lattice spin glasses and the 4D Edwards-Anderson model. Work in collaboration with Miguel Aguilar-Janita, Victor Martin-Mayor, Javier Moreno-Gordo, Giorgio Parisi, Federico Ricci-Tersenghi, Juan J. Ruiz-Lorenzo

To improve the storage capacity of the Hopfield model, we develop a version of the dreaming algorithm that is perpetually exposed to data and therefore called Daydreaming. Daydreaming is not destructive and converges asymptotically to a stationary coupling matrix. When trained on random uncorrelated examples, the model shows optimal performance in terms of the size of the basins of attraction of stored examples and the quality of reconstruction. We also train the Daydreaming algorithm on correlated data obtained via the random-features model and argue that it spontaneously exploits the correlations thus increasing even further the storage capacity and the size of the basins of attraction. Moreover, the Daydreaming algorithm is also able to stabilize the features hidden in the data. Finally, we test Daydreaming on the MNIST dataset and show that it still works surprisingly well, producing attractors that are close to unseen examples and class prototypes.