404 research outputs found
Free zero-range processes on networks
A free zero-range process (FRZP) is a simple stochastic process describing
the dynamics of a gas of particles hopping between neighboring nodes of a
network. We discuss three different cases of increasing complexity: (a) FZRP on
a rigid geometry where the network is fixed during the process, (b) FZRP on a
random graph chosen from a given ensemble of networks, (c) FZRP on a dynamical
network whose topology continuously changes during the process in a way which
depends on the current distribution of particles. The case (a) provides a very
simple realization of the phenomenon of condensation which manifests as the
appearance of a condensate of particles on the node with maximal degree. The
case (b) is very interesting since the averaging over typical ensembles of
graphs acts as a kind of homogenization of the system which makes all nodes
identical from the point of view of the FZRP. In the case (c), the distribution
of particles and the dynamics of network are coupled to each other. The
strength of this coupling depends on the ratio of two time scales: for changes
of the topology and of the FZRP. We will discuss a specific example of that
type of interaction and show that it leads to an interesting phase diagram.Comment: 11 pages, 4 figures, to appear in Proceedings of SPIE Symposium
"Fluctuations and Noise 2007", Florence, 20-24 May 200
Balls-in-boxes condensation on networks
We discuss two different regimes of condensate formation in zero-range
processes on networks: on a q-regular network, where the condensate is formed
as a result of a spontaneous symmetry breaking, and on an irregular network,
where the symmetry of the partition function is explicitly broken. In the
latter case we consider a minimal irregularity of the q-regular network
introduced by a single Q-node with degree Q>q. The statics and dynamics of the
condensation depends on the parameter log(Q/q), which controls the exponential
fall-off of the distribution of particles on regular nodes and the typical time
scale for melting of the condensate on the Q-node which increases exponentially
with the system size . This behavior is different than that on a q-regular
network where log(Q/q)=0 and where the condensation results from the
spontaneous symmetry breaking of the partition function, which is invariant
under a permutation of particle occupation numbers on the q-nodes of the
network. In this case the typical time scale for condensate melting is known to
increase typically as a power of the system size.Comment: 7 pages, 3 figures, submitted to the "Chaos" focus issue on
"Optimization in Networks" (scheduled to appear as Volume 17, No. 2, 2007
Quantum widening of CDT universe
The physical phase of Causal Dynamical Triangulations (CDT) is known to be
described by an effective, one-dimensional action in which three-volumes of the
underlying foliation of the full CDT play a role of the sole degrees of
freedom. Here we map this effective description onto a statistical-physics
model of particles distributed on 1d lattice, with site occupation numbers
corresponding to the three-volumes. We identify the emergence of the quantum
de-Sitter universe observed in CDT with the condensation transition known from
similar statistical models. Our model correctly reproduces the shape of the
quantum universe and allows us to analytically determine quantum corrections to
the size of the universe. We also investigate the phase structure of the model
and show that it reproduces all three phases observed in computer simulations
of CDT. In addition, we predict that two other phases may exists, depending on
the exact form of the discretised effective action and boundary conditions. We
calculate various quantities such as the distribution of three-volumes in our
model and discuss how they can be compared with CDT.Comment: 19 pages, 13 figure
Spectrum of the Dirac operator coupled to two-dimensional quantum gravity
We implement fermions on dynamical random triangulation and determine
numerically the spectrum of the Dirac-Wilson operator D for the system of
Majorana fermions coupled to two-dimensional Euclidean quantum gravity. We
study the dependence of the spectrum of the operator (epsilon D) on the hopping
parameter. We find that the distributions of the lowest eigenvalues become
discrete when the hopping parameter approaches the value 1/sqrt{3}. We show
that this phenomenon is related to the behavior of the system in the
'antiferromagnetic' phase of the corresponding Ising model. Using finite size
analysis we determine critical exponents controlling the scaling of the lowest
eigenvalue of the spectrum including the Hausdorff dimension d_H and the
exponent kappa which tells us how fast the pseudo-critical value of the hopping
parameter approaches its infinite volume limit.Comment: 26 pages, Latex + 23 eps figs, extended analysis of the spectrum,
added figure
Zero-range process with long-range interactions at a T-junction
A generalized zero-range process with a limited number of long-range
interactions is studied as an example of a transport process in which particles
at a T-junction make a choice of which branch to take based on traffic levels
on each branch. The system is analysed with a self-consistent mean-field
approximation which allows phase diagrams to be constructed. Agreement between
the analysis and simulations is found to be very good.Comment: 21 pages, 6 figure
Pair-factorized steady states on arbitrary graphs
Stochastic mass transport models are usually described by specifying hopping
rates of particles between sites of a given lattice, and the goal is to predict
the existence and properties of the steady state. Here we ask the reverse
question: given a stationary state that factorizes over links (pairs of sites)
of an arbitrary connected graph, what are possible hopping rates that converge
to this state? We define a class of hopping functions which lead to the same
steady state and guarantee current conservation but may differ by the induced
current strength. For the special case of anisotropic hopping in two dimensions
we discuss some aspects of the phase structure. We also show how this case can
be traced back to an effective zero-range process in one dimension which is
solvable for a large class of hopping functions.Comment: IOP style, 9 pages, 1 figur
Fermions in three-dimensional spinfoam quantum gravity
We study the coupling of massive fermions to the quantum mechanical dynamics
of spacetime emerging from the spinfoam approach in three dimensions. We first
recall the classical theory before constructing a spinfoam model of quantum
gravity coupled to spinors. The technique used is based on a finite expansion
in inverse fermion masses leading to the computation of the vacuum to vacuum
transition amplitude of the theory. The path integral is derived as a sum over
closed fermionic loops wrapping around the spinfoam. The effects of quantum
torsion are realised as a modification of the intertwining operators assigned
to the edges of the two-complex, in accordance with loop quantum gravity. The
creation of non-trivial curvature is modelled by a modification of the pure
gravity vertex amplitudes. The appendix contains a review of the geometrical
and algebraic structures underlying the classical coupling of fermions to three
dimensional gravity.Comment: 40 pages, 3 figures, version accepted for publication in GER
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