Multi-species mean-field spin-glasses. Rigorous results

We study a multi-species spin glass system where the density of each species is kept fixed at increasing volumes. The model reduces to the Sherrington-Kirkpatrick one for the single species case. The existence of the thermodynamic limit is proved for all densities values under a convexity condition on the interaction. The thermodynamic properties of the model are investigated and the annealed, the replica symmetric and the replica symmetry breaking bounds are proved using Guerra's scheme. The annealed approximation is proved to be exact under a high temperature condition. We show that the replica symmetric solution has negative entropy at low temperatures. We study the properties of a suitably defined replica symmetry breaking solution and we optimise it within a ziggurat ansatz. The generalized order parameter is described by a Parisi-like partial differential equation.Comment: 17 pages, to appear in Annales Henri Poincar\`

Bayesian analysis of latent variable models in finance

Lucas, A. [Promotor]Koopman, S.J. [Promotor]Hoogerheide, L.F. [Copromotor

Log-periodic drift oscillations in self-similar billiards

We study a particle moving at unit speed in a self-similar Lorentz billiard channel; the latter consists of an infinite sequence of cells which are identical in shape but growing exponentially in size, from left to right. We present numerical computation of the drift term in this system and establish the logarithmic periodicity of the corrections to the average drift

Criticality in diluted ferromagnet

We perform a detailed study of the critical behavior of the mean field diluted Ising ferromagnet by analytical and numerical tools. We obtain self-averaging for the magnetization and write down an expansion for the free energy close to the critical line. The scaling of the magnetization is also rigorously obtained and compared with extensive Monte Carlo simulations. We explain the transition from an ergodic region to a non trivial phase by commutativity breaking of the infinite volume limit and a suitable vanishing field. We find full agreement among theory, simulations and previous results.Comment: 23 pages, 3 figure

Conformal Dynamics of Precursors to Fracture

An exact integro-differential equation for the conformal map from the unit circle to the boundary of an evolving cavity in a stressed 2-dimensional solid is derived. This equation provides an accurate description of the dynamics of precursors to fracture when surface diffusion is important. The solution predicts the creation of sharp grooves that eventually lead to material failure via rapid fracture. Solutions of the new equation are demonstrated for the dynamics of an elliptical cavity and the stability of a circular cavity under biaxial stress, including the effects of surface stress.Comment: 4 pages, 3 figure

Ferromagnetic models for cooperative behavior: Revisiting Universality in complex phenomena

Ferromagnetic models are harmonic oscillators in statistical mechanics. Beyond their original scope in tackling phase transition and symmetry breaking in theoretical physics, they are nowadays experiencing a renewal applicative interest as they capture the main features of disparate complex phenomena, whose quantitative investigation in the past were forbidden due to data lacking. After a streamlined introduction to these models, suitably embedded on random graphs, aim of the present paper is to show their importance in a plethora of widespread research fields, so to highlight the unifying framework reached by using statistical mechanics as a tool for their investigation. Specifically we will deal with examples stemmed from sociology, chemistry, cybernetics (electronics) and biology (immunology).Comment: Contributing to the proceedings of the Conference "Mathematical models and methods for Planet Heart", INdAM, Rome 201

Drift of particles in self-similar systems and its Liouvillian interpretation

We study the dynamics of classical particles in different classes of spatially extended self-similar systems, consisting of (i) a self-similar Lorentz billiard channel, (ii) a self-similar graph, and (iii) a master equation. In all three systems the particles typically drift at constant velocity and spread ballistically. These transport properties are analyzed in terms of the spectral properties of the operator evolving the probability densities. For systems (i) and (ii), we explain the drift from the properties of the Pollicott-Ruelle resonance spectrum and corresponding eigenvectorsComment: To appear in Phys. Rev.

Analogue neural networks on correlated random graphs

We consider a generalization of the Hopfield model, where the entries of patterns are Gaussian and diluted. We focus on the high-storage regime and we investigate analytically the topological properties of the emergent network, as well as the thermodynamic properties of the model. We find that, by properly tuning the dilution in the pattern entries, the network can recover different topological regimes characterized by peculiar scalings of the average coordination number with respect to the system size. The structure is also shown to exhibit a large degree of cliquishness, even when very sparse. Moreover, we obtain explicitly the replica symmetric free energy and the self-consistency equations for the overlaps (order parameters of the theory), which turn out to be classical weighted sums of 'sub-overlaps' defined on all possible sub-graphs. Finally, a study of criticality is performed through a small-overlap expansion of the self-consistencies and through a whole fluctuation theory developed for their rescaled correlations: Both approaches show that the net effect of dilution in pattern entries is to rescale the critical noise level at which ergodicity breaks down.Comment: 34 pages, 3 figure

Equilibrium statistical mechanics on correlated random graphs

Biological and social networks have recently attracted enormous attention between physicists. Among several, two main aspects may be stressed: A non trivial topology of the graph describing the mutual interactions between agents exists and/or, typically, such interactions are essentially (weighted) imitative. Despite such aspects are widely accepted and empirically confirmed, the schemes currently exploited in order to generate the expected topology are based on a-priori assumptions and in most cases still implement constant intensities for links. Here we propose a simple shift in the definition of patterns in an Hopfield model to convert frustration into dilution: By varying the bias of the pattern distribution, the network topology -which is generated by the reciprocal affinities among agents - crosses various well known regimes (fully connected, linearly diverging connectivity, extreme dilution scenario, no network), coupled with small world properties, which, in this context, are emergent and no longer imposed a-priori. The model is investigated at first focusing on these topological properties of the emergent network, then its thermodynamics is analytically solved (at a replica symmetric level) by extending the double stochastic stability technique, and presented together with its fluctuation theory for a picture of criticality. At least at equilibrium, dilution simply decreases the strength of the coupling felt by the spins, but leaves the paramagnetic/ferromagnetic flavors unchanged. The main difference with respect to previous investigations and a naive picture is that within our approach replicas do not appear: instead of (multi)-overlaps as order parameters, we introduce a class of magnetizations on all the possible sub-graphs belonging to the main one investigated: As a consequence, for these objects a closure for a self-consistent relation is achieved.Comment: 30 pages, 4 figure

Steady-state conduction in self-similar billiards

The self-similar Lorentz billiard channel is a spatially extended deterministic dynamical system which consists of an infinite one-dimensional sequence of cells whose sizes increase monotonically according to their indices. This special geometry induces a nonequilibrium stationary state with particles flowing steadily from the small to the large scales. The corresponding invariant measure has fractal properties reflected by the phase-space contraction rate of the dynamics restricted to a single cell with appropriate boundary conditions. In the near-equilibrium limit, we find numerical agreement between this quantity and the entropy production rate as specified by thermodynamics