90 research outputs found
GEMs and amplitude bounds in the colored Boulatov model
In this paper we construct a methodology for separating the divergencies due
to different topological manifolds dual to Feynman graphs in colored group
field theory. After having introduced the amplitude bounds using propagator
cuts, we show how Graph-Encoded-Manifolds (GEM) techniques can be used in order
to factorize divergencies related to different parts of the dual topologies of
the Feynman graphs in the general case. We show the potential of the formalism
in the case of 3-dimensional solid torii in the colored Boulatov model.Comment: 20 pages; 20 Figures; Style changed, discussion improved and typos
corrected, citations added; These GEMs are not related to "Global Embedding
Minkowskian spacetimes
Quantum quenches and thermalization on scale-free graphs
We show that after a quantum quench of the parameter controlling the number
of particles in a Fermi-Hubbard model on scale free graphs, the distribution of
energy modes follows a power law dependent on the quenched parameter and the
connectivity of the graph. This paper contributes to the literature of quantum
quenches on lattices, in which, for many integrable lattice models the
distribution of modes after a quench thermalizes to a Generalized Gibbs
Ensemble; this paper provides another example of distribution which can arise
after relaxation. We argue that the main role is played by the symmetry of the
underlying lattice which, in the case we study, is scale free, and to the
distortion in the density of modes.Comment: 10 pages; 5 figures; accepted for publication in JTA
Asymptotic behavior of memristive circuits
The interest in memristors has risen due to their possible application both
as memory units and as computational devices in combination with CMOS. This is
in part due to their nonlinear dynamics, and a strong dependence on the circuit
topology. We provide evidence that also purely memristive circuits can be
employed for computational purposes. In the present paper we show that a
polynomial Lyapunov function in the memory parameters exists for the case of DC
controlled memristors. Such Lyapunov function can be asymptotically
approximated with binary variables, and mapped to quadratic combinatorial
optimization problems. This also shows a direct parallel between memristive
circuits and the Hopfield-Little model. In the case of Erdos-Renyi random
circuits, we show numerically that the distribution of the matrix elements of
the projectors can be roughly approximated with a Gaussian distribution, and
that it scales with the inverse square root of the number of elements. This
provides an approximated but direct connection with the physics of disordered
system and, in particular, of mean field spin glasses. Using this and the fact
that the interaction is controlled by a projector operator on the loop space of
the circuit. We estimate the number of stationary points of the approximate
Lyapunov function and provide a scaling formula as an upper bound in terms of
the circuit topology only.Comment: 20 pages, 8 figures; proofs corrected, figures changed; results
substantially unchanged; to appear in Entrop
Trajectories entropy in dynamical graphs with memory
In this paper we investigate the application of non-local graph entropy to
evolving and dynamical graphs. The measure is based upon the notion of Markov
diffusion on a graph, and relies on the entropy applied to trajectories
originating at a specific node. In particular, we study the model of
reinforcement-decay graph dynamics, which leads to scale free graphs. We find
that the node entropy characterizes the structure of the network in the two
parameter phase-space describing the dynamical evolution of the weighted graph.
We then apply an adapted version of the entropy measure to purely memristive
circuits. We provide evidence that meanwhile in the case of DC voltage the
entropy based on the forward probability is enough to characterize the graph
properties, in the case of AC voltage generators one needs to consider both
forward and backward based transition probabilities. We provide also evidence
that the entropy highlights the self-organizing properties of memristive
circuits, which re-organizes itself to satisfy the symmetries of the underlying
graph.Comment: 15 pages one column, 10 figures; new analysis and memristor models
added. Text improve
Bounds on transient instability for complex ecosystems
Stability is a desirable property of complex ecosystems. If a community of
interacting species is at a stable equilibrium point then it is able to
withstand small perturbations to component species' abundances without
suffering adverse effects. In ecology, the Jacobian matrix evaluated at an
equilibrium point is known as the community matrix, which describes the
population dynamics of interacting species. A system's asymptotic short- and
long-term behaviour can be determined from eigenvalues derived from the
community matrix. Here we use results from the theory of pseudospectra to
describe intermediate, transient dynamics. We first recover the established
result that the transition from stable to unstable dynamics includes a region
of `transient instability', where the effect of a small perturbation to
species' abundances---to the population vector---is amplified before ultimately
decaying. Then we show that the shift from stability to transient instability
can be affected by uncertainty in, or small changes to, entries in the
community matrix, and determine lower and upper bounds to the maximum amplitude
of perturbations to the population vector. Of five different types of community
matrix, we find that amplification is least severe when predator-prey
interactions dominate. This analysis is relevant to other systems whose
dynamics can be expressed in terms of the Jacobian matrix. Our results will
lead to improved understanding of how multiple perturbations to a complex
system may irrecoverably break stability.Comment: 7 pages, two columns, 3 figures; text improved - Accepted for
publication on PLoS On
Properties of Quantum Graphity at Low Temperature
We present a mapping of dynamical graphs and, in particular, the graphs used
in the Quantum Graphity models for emergent geometry, into an Ising hamiltonian
on the line graph of a complete graph with a fixed number of vertices. We use
this method to study the properties of Quantum Graphity models at low
temperature in the limit in which the valence coupling constant of the model is
much greater than the coupling constants of the loop terms. Using mean field
theory we find that an order parameter for the model is the average valence of
the graph. We calculate the equilibrium distribution for the valence as an
implicit function of the temperature. In the approximation in which the
temperature is low, we find the first two Taylor coefficients of the valence in
the temperature expansion. A discussion of the susceptibility function and a
generalization of the model are given in the end.Comment: 21 pages, 5 figures; a plot added, corrections added; Accepted for
publication in PR
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