2,795 research outputs found
Infinite cycles in the random stirring model on trees
We prove that, in the random stirring model of parameter T on an infinite
rooted tree each of whose vertices has at least two offspring, infinite cycles
exist almost surely, provided that T is sufficiently high.
In the appendices, the bound on degree above which the result holds is
improved slightly.Comment: 23 pages, two figure
Sharp phase transition in the random stirring model on trees
We establish that the phase transition for infinite cycles in the random
stirring model on an infinite regular tree of high degree is sharp. That is, we
prove that there exists d_0 such that, for any d \geq d_0, the set of parameter
values at which the random stirring model on the rooted regular tree with
offspring degree d almost surely contains an infinite cycle consists of a
semi-infinite interval. The critical point at the left-hand end of this
interval is at least 1/d + 1/(2d^2) and at most 1/d + 2/(d^2).
This version is a major revision, with a much shorter proof. Principal among
the changes are a reworking of the argument in Section 4 of the old version,
which was proposed by a referee, and the use of a simpler means of handling a
boundary case, which eliminates the previous Section 6.Comment: 20 pages, three figures. A short explanation of Proposition 3.2 has
been added. Probab. Theory and Related Fields, to appea
Metastability of Queuing Networks with Mobile Servers
We study symmetric queuing networks with moving servers and FIFO service
discipline. The mean-field limit dynamics demonstrates unexpected behavior
which we attribute to the meta-stability phenomenon. Large enough finite
symmetric networks on regular graphs are proved to be transient for arbitrarily
small inflow rates. However, the limiting non-linear Markov process possesses
at least two stationary solutions. The proof of transience is based on
martingale techniques
Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks
This article is a mini-review about electrical current flows in networks from
the perspective of statistical physics. We briefly discuss analytical methods
to solve the conductance of an arbitrary resistor network. We then turn to
basic results related to percolation: namely, the conduction properties of a
large random resistor network as the fraction of resistors is varied. We focus
on how the conductance of such a network vanishes as the percolation threshold
is approached from above. We also discuss the more microscopic current
distribution within each resistor of a large network. At the percolation
threshold, this distribution is multifractal in that all moments of this
distribution have independent scaling properties. We will discuss the meaning
of multifractal scaling and its implications for current flows in networks,
especially the largest current in the network. Finally, we discuss the relation
between resistor networks and random walks and show how the classic phenomena
of recurrence and transience of random walks are simply related to the
conductance of a corresponding electrical network.Comment: 27 pages & 10 figures; review article for the Encyclopedia of
Complexity and System Science (Springer Science
Spectral Properties of Quantum Walks on Rooted Binary Trees
We define coined Quantum Walks on the infinite rooted binary tree given by
unitary operators on an associated infinite dimensional Hilbert space,
depending on a unitary coin matrix , and study their spectral
properties. For circulant unitary coin matrices , we derive an equation for
the Carath\'eodory function associated to the spectral measure of a cyclic
vector for . This allows us to show that for all circulant unitary coin
matrices, the spectrum of the Quantum Walk has no singular continuous
component. Furthermore, for coin matrices which are orthogonal circulant
matrices, we show that the spectrum of the Quantum Walk is absolutely
continuous, except for four coin matrices for which the spectrum of is
pure point
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