8,113 research outputs found
Growth Series and Random Walks on Some Hyperbolic Graphs
Consider the tesselation of the hyperbolic plane by m-gons, l per vertex. In
its 1-skeleton, we compute the growth series of vertices, geodesics, tuples of
geodesics with common extremities. We also introduce and enumerate "holly
trees", a family of reduced loops in these graphs. We then apply Grigorchuk's
result relating cogrowth and random walks to obtain lower estimates on the
spectral radius of the Markov operator associated with a symmetric random walk
on these graphs.Comment: 21 pages. to appear in monash. mat
Scattering theory and discrete-time quantum walks
We study quantum walks on general graphs from the point of view of scattering
theory. For a general finite graph we choose two vertices and attach one half
line to each. We are interested in walks that proceed from one half line,
through the graph, to the other. The particle propagates freely on the half
lines but is scattered at each vertex in the original graph. The probability of
starting on one line and reaching the other after n steps can be expressed in
terms of the transmission amplitude for the graph. An example is presented.Comment: 7 pages, Latex, replaced with published versio
Quantum walks based on an interferometric analogy
There are presently two models for quantum walks on graphs. The "coined" walk
uses discrete time steps, and contains, besides the particle making the walk, a
second quantum system, the coin, that determines the direction in which the
particle will move. The continuous walk operates with continuous time. Here a
third model for a quantum walk is proposed, which is based on an analogy to
optical interferometers. It is a discrete-time model, and the unitary operator
that advances the walk one step depends only on the local structure of the
graph on which the walk is taking place. No quantum coin is introduced. This
type of walk allows us to introduce elements, such as phase shifters, that have
no counterpart in classical random walks. Walks on the line and cycle are
discussed in some detail, and a probability current for these walks is
introduced. The relation to the coined quantum walk is also discussed. The
paper concludes by showing how to define these walks for a general graph.Comment: Latex,18 pages, 5 figure
Quantum and classical localisation and the Manhattan lattice
We consider a network model, embedded on the Manhattan lattice, of a quantum
localisation problem belonging to symmetry class C. This arises in the context
of quasiparticle dynamics in disordered spin-singlet superconductors which are
invariant under spin rotations but not under time reversal. A mapping exists
between problems belonging to this symmetry class and certain classical random
walks which are self-avoiding and have attractive interactions; we exploit this
equivalence, using a study of the classical random walks to gain information
about the corresponding quantum problem. In a field-theoretic approach, we show
that the interactions may flow to one of two possible strong coupling regimes
separated by a transition: however, using Monte Carlo simulations we show that
the walks are in fact always compact two-dimensional objects with a
well-defined one-dimensional surface, indicating that the corresponding quantum
system is localised.Comment: 11 pages, 8 figure
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