Fluctuations induce transitions in frustrated sparse networks

We analyze, by means of statistical mechanics, a sparse network with random competitive interactions among dichotomic variables pasted on the nodes, namely a Viana-Bray model. The model is described by an infinite series of order parameters (the multi-overlaps) and has two tunable degrees of freedom: the noise level and the connectivity (the averaged number of links). We show that there are no multiple transition lines, one for every order parameter, as a naive approach would suggest, but just one corresponding to ergodicity breaking. We explain this scenario within a novel and simple mathematical technique via a driving mechanism such that, as the first order parameter (the two replica overlap) becomes different from zero due to a real second order phase transition (with properly associated diverging rescaled fluctuations), it enforces all the other multi-overlaps toward positive values thanks to the strong correlations which develop among themselves and the two replica overlap at the critical line

A numerical investigation of the jamming transition in traffic flow on diluted planar networks

In order to develop a toy model for car's traffic in cities, in this paper we analyze, by means of numerical simulations, the transition among fluid regimes and a congested jammed phase of the flow of "kinetically constrained" hard spheres in planar random networks similar to urban roads. In order to explore as timescales as possible, at a microscopic level we implement an event driven dynamics as the infinite time limit of a class of already existing model (e.g. "Follow the Leader") on an Erdos-Renyi two dimensional graph, the crossroads being accounted by standard Kirchoff density conservations. We define a dynamical order parameter as the ratio among the moving spheres versus the total number and by varying two control parameters (density of the spheres and coordination number of the network) we study the phase transition. At a mesoscopic level it respects an, again suitable adapted, version of the Lighthill-Whitham model, which belongs to the fluid-dynamical approach to the problem. At a macroscopic level the model seems to display a continuous transition from a fluid phase to a jammed phase when varying the density of the spheres (the amount of cars in a city-like scenario) and a discontinuous jump when varying the connectivity of the underlying network.Comment: accepted in Int.J.Mod.Phys.

Exact solution of the van der Waals model in the critical region

The celebrated van der Waals model describes simple fluids in the thermodynamic limit and predicts the existence of a critical point associated to the gas-liquid phase transition. However the behaviour of critical isotherms according to the equation of state, where a gas-liquid phase transition occurs, significantly departs from experimental observations. The correct critical isotherms are heuristically re-established via the Maxwell equal areas rule. A long standing open problem in mean field theory is concerned with the analytic description of van der Waals isotherms for a finite size system that is consistent, in the thermodynamic limit, with the Maxwell prescription. Inspired by the theory of nonlinear conservation laws, we propose a novel mean field approach, based on statistical mechanics, that allows to calculate the van der Waals partition function for a system of large but finite number of particles $N$. Our partition function naturally extends to the whole space of thermodynamic variables, reproduces, in the thermodynamic limit $N\to \infty$, the classical results outside the critical region and automatically encodes Maxwell's prescription. We show that isothermal curves evolve in the space of thermodynamic variables like nonlinear breaking waves and the criticality is explained as the mechanism of formation of a classical hydrodynamic shock

Overlap Fluctuations from Random Overlap Structures

We investigate overlap fluctuations of the Sherrington-Kirkpatrick mean field spin glass model in the framework of the Random Over- lap Structure (ROSt). The concept of ROSt has been introduced recently by Aizenman and coworkers, who developed a variational approach to the Sherrington-Kirkpatrick model. We propose here an iterative procedure to show that, in the so-called Boltzmann ROSt, Aizenman-Contucci (AC) polynomials naturally arise for almost all values of the inverse temperature (not in average over some interval only). The same results can be obtained in any ROSt, including therefore the Parisi structure. The AC polynomials impose restric- tions on the overlap fluctuations in agreement with Parisi theory.Comment: 18 page

Free energies of Boltzmann Machines: self-averaging, annealed and replica symmetric approximations in the thermodynamic limit

Restricted Boltzmann machines (RBMs) constitute one of the main models for machine statistical inference and they are widely employed in Artificial Intelligence as powerful tools for (deep) learning. However, in contrast with countless remarkable practical successes, their mathematical formalization has been largely elusive: from a statistical-mechanics perspective these systems display the same (random) Gibbs measure of bi-partite spin-glasses, whose rigorous treatment is notoriously difficult. In this work, beyond providing a brief review on RBMs from both the learning and the retrieval perspectives, we aim to contribute to their analytical investigation, by considering two distinct realizations of their weights (i.e., Boolean and Gaussian) and studying the properties of their related free energies. More precisely, focusing on a RBM characterized by digital couplings, we first extend the Pastur-Shcherbina-Tirozzi method (originally developed for the Hopfield model) to prove the self-averaging property for the free energy, over its quenched expectation, in the infinite volume limit, then we explicitly calculate its simplest approximation, namely its annealed bound. Next, focusing on a RBM characterized by analogical weights, we extend Guerra's interpolating scheme to obtain a control of the quenched free-energy under the assumption of replica symmetry: we get self-consistencies for the order parameters (in full agreement with the existing Literature) as well as the critical line for ergodicity breaking that turns out to be the same obtained in AGS theory. As we discuss, this analogy stems from the slow-noise universality. Finally, glancing beyond replica symmetry, we analyze the fluctuations of the overlaps for an estimate of the (slow) noise affecting the retrieval of the signal, and by a stability analysis we recover the Aizenman-Contucci identities typical of glassy systems.Comment: 21 pages, 1 figur