53 research outputs found
Macroscopic limit of a bipartite Curie-Weiss model: a dynamical approach
We analyze the Glauber dynamics for a bi-populated Curie-Weiss model. We
obtain the limiting behavior of the empirical averages in the limit of
infinitely many particles. We then characterize the phase space of the model in
absence of magnetic field and we show that several phase transitions in the
inter-groups interaction strength occur.Comment: 18 pages, 3 figure
The role of disorder in the dynamics of critical fluctuations of mean field models
The purpose of this paper is to analyze how the disorder affects the dynamics
of critical fluctuations for two different types of interacting particle
system: the Curie-Weiss and Kuramoto model. The models under consideration are
a collection of spins and rotators respectively. They both are subject to a
mean field interaction and embedded in a site-dependent, i.i.d. random
environment. As the number of particles goes to infinity their limiting
dynamics become deterministic and exhibit phase transition. The main result
concern the fluctuations around this deterministic limit at the critical point
in the thermodynamic limit. From a qualitative point of view, it indicates that
when disorder is added spin and rotator systems belong to two different classes
of universality, which is not the case for the homogeneous models (i.e.,
without disorder).Comment: 41 page
Free completely random measures
Free probability is a noncommutative probability theory introduced by Voiculescu where the concept of independence of classical probability is replaced by the concept of freeness. An important connection between free and classical infinite divisibility was established by Bercovici and Pata (1999) in form of a bijection, mapping the class of classical infinitely divisible laws into the class of free infinitely divisible laws. A particular class of infinitely divisible laws are the completely random measures introduced by Kingman (1967). In this paper, a free analogous of completely random measures is introduced and, a free Poisson process characterization is provided as well as a representation through a free cumulant transform. Furthermore, some examples are displayed.Bayesian non parametrics, Bercovici-Pata bijection, Free completely random measures, Free infinite divisibility, Free probability
Collective periodicity in mean-field models of cooperative behavior
We propose a way to break symmetry in stochastic dynamics by introducing a
dissipation term. We show in a specific mean-field model, that if the
reversible model undergoes a phase transition of ferromagnetic type, then its
dissipative counterpart exhibits periodic orbits in the thermodynamic limit.Comment: 19 pages, 3 figure
Free completely random measures
Free probability is a noncommutative probability theory introduced by Voiculescu
where the concept of independence
of classical probability is replaced by the concept of freeness. An important connection
between free and classical
infinite divisibility was established by Bercovici and Pata (1999) in form of a bijection,
mapping the class of classical
infinitely divisible laws into the class of free infinitely divisible laws.
A particular class of infinitely divisible laws are the completely random measures
introduced by Kingman (1967). In
this paper, a free analogous of completely random measures is introduced and, a free
Poisson process characterization
is provided as well as a representation through a free cumulant transform. Furthermore,
some examples are displayed
Synchronization and Spin-Flop Transitions for a Mean-Field XY Model in Random Field
We characterize the phase space for the infinite volume limit of a ferromagnetic mean-field XY model in a random field pointing in one direction with two symmetric values. We determine the stationary solutions and detect possible phase transitions in the interaction strength for fixed random field intensity. We show that at low temperature magnetic ordering appears perpendicularly to the field. The latter situation corresponds to a spin-flop transition
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