1,026 research outputs found
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
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