58 research outputs found
Anderson model on Bethe lattices: density of states, localization properties and isolated eigenvalue
We revisit the Anderson localization problem on Bethe lattices, putting in
contact various aspects which have been previously only discussed separately.
For the case of connectivity 3 we compute by the cavity method the density of
states and the evolution of the mobility edge with disorder. Furthermore, we
show that below a certain critical value of the disorder the smallest
eigenvalue remains delocalized and separated by all the others (localized) ones
by a gap. We also study the evolution of the mobility edge at the center of the
band with the connectivity, and discuss the large connectivity limit.Comment: 13 pages, 4 figures, Proceedings of the YKIS2009 conference,
references adde
Recovery thresholds in the sparse planted matching problem
We consider the statistical inference problem of recovering an unknown
perfect matching, hidden in a weighted random graph, by exploiting the
information arising from the use of two different distributions for the weights
on the edges inside and outside the planted matching. A recent work has
demonstrated the existence of a phase transition, in the large size limit,
between a full and a partial recovery phase for a specific form of the weights
distribution on fully connected graphs. We generalize and extend this result in
two directions: we obtain a criterion for the location of the phase transition
for generic weights distributions and possibly sparse graphs, exploiting a
technical connection with branching random walk processes, as well as a
quantitatively more precise description of the critical regime around the phase
transition.Comment: 19 pages, 8 figure
Biased landscapes for random Constraint Satisfaction Problems
The typical complexity of Constraint Satisfaction Problems (CSPs) can be
investigated by means of random ensembles of instances. The latter exhibit many
threshold phenomena besides their satisfiability phase transition, in
particular a clustering or dynamic phase transition (related to the tree
reconstruction problem) at which their typical solutions shatter into
disconnected components. In this paper we study the evolution of this
phenomenon under a bias that breaks the uniformity among solutions of one CSP
instance, concentrating on the bicoloring of k-uniform random hypergraphs. We
show that for small k the clustering transition can be delayed in this way to
higher density of constraints, and that this strategy has a positive impact on
the performances of Simulated Annealing algorithms. We characterize the modest
gain that can be expected in the large k limit from the simple implementation
of the biasing idea studied here. This paper contains also a contribution of a
more methodological nature, made of a review and extension of the methods to
determine numerically the discontinuous dynamic transition threshold.Comment: 32 pages, 16 figure
On the stochastic dynamics of disordered spin models
In this article we discuss several aspects of the stochastic dynamics of spin
models. The paper has two independent parts. Firstly, we explore a few
properties of the multi-point correlations and responses of generic systems
evolving in equilibrium with a thermal bath. We propose a fluctuation principle
that allows us to derive fluctuation-dissipation relations for many-time
correlations and linear responses. We also speculate on how these features will
be modified in systems evolving slowly out of equilibrium, as
finite-dimensional or dilute spin-glasses. Secondly, we present a formalism
that allows one to derive a series of approximated equations that determine the
dynamics of disordered spin models on random (hyper) graphs.Comment: 25 page
The asymptotics of the clustering transition for random constraint satisfaction problems
Random Constraint Satisfaction Problems exhibit several phase transitions
when their density of constraints is varied. One of these threshold phenomena,
known as the clustering or dynamic transition, corresponds to a transition for
an information theoretic problem called tree reconstruction. In this article we
study this threshold for two CSPs, namely the bicoloring of -uniform
hypergraphs with a density of constraints, and the -coloring of
random graphs with average degree . We show that in the large limit
the clustering transition occurs for , , where is the same constant for both models. We
characterize via a functional equation, solve the latter
numerically to estimate , and obtain an analytic
lowerbound . Our
analysis unveils a subtle interplay of the clustering transition with the
rigidity (naive reconstruction) threshold that occurs on the same asymptotic
scale at .Comment: 35 pages, 8 figure
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