25,780 research outputs found
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
A Generalized Epidemic Process and Tricritical Dynamic Percolation
The renowned general epidemic process describes the stochastic evolution of a
population of individuals which are either susceptible, infected or dead. A
second order phase transition belonging to the universality class of dynamic
isotropic percolation lies between endemic or pandemic behavior of the process.
We generalize the general epidemic process by introducing a fourth kind of
individuals, viz. individuals which are weakened by the process but not yet
infected. This sensibilization gives rise to a mechanism that introduces a
global instability in the spreading of the process and therefore opens the
possibility of a discontinuous transition in addition to the usual continuous
percolation transition. The tricritical point separating the lines of first and
second order transitions constitutes a new universality class, namely the
universality class of tricritical dynamic isotropic percolation. Using
renormalized field theory we work out a detailed scaling description of this
universality class. We calculate the scaling exponents in an
-expansion below the upper critical dimension for various
observables describing tricritical percolation clusters and their spreading
properties. In a remarkable contrast to the usual percolation transition, the
exponents and governing the two order parameters,
viz. the mean density and the percolation probability, turn out to be different
at the tricritical point. In addition to the scaling exponents we calculate for
all our static and dynamic observables logarithmic corrections to the
mean-field scaling behavior at .Comment: 21 pages, 10 figures, version to appear in Phys. Rev.
Balanced Symmetric Functions over
Under mild conditions on , we give a lower bound on the number of
-variable balanced symmetric polynomials over finite fields , where
is a prime number. The existence of nonlinear balanced symmetric
polynomials is an immediate corollary of this bound. Furthermore, we conjecture
that are the only nonlinear balanced elementary symmetric
polynomials over GF(2), where , and we prove various results in support of this conjecture.Comment: 21 page
Unwrapping phase fluctuations in one dimension
Correlation functions in one-dimensional complex scalar field theory provide
a toy model for phase fluctuations, sign problems, and signal-to-noise problems
in lattice field theory. Phase unwrapping techniques from signal processing are
applied to lattice field theory in order to map compact random phases to
noncompact random variables that can be numerically sampled without sign or
signal-to-noise problems. A cumulant expansion can be used to reconstruct
average correlation functions from moments of unwrapped phases, but points
where the field magnitude fluctuates close to zero lead to ambiguities in the
definition of the unwrapped phase and significant noise at higher orders in the
cumulant expansion. Phase unwrapping algorithms that average fluctuations over
physical length scales improve, but do not completely resolve, these issues in
one dimension. Similar issues are seen in other applications of phase
unwrapping, where they are found to be more tractable in higher dimensions.Comment: 14 pages, 7 figures. arXiv admin note: text overlap with
arXiv:1806.0183
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