32,072 research outputs found
Deterministic Random Walks on Regular Trees
Jim Propp's rotor router model is a deterministic analogue of a random walk
on a graph. Instead of distributing chips randomly, each vertex serves its
neighbors in a fixed order.
Cooper and Spencer (Comb. Probab. Comput. (2006)) show a remarkable
similarity of both models. If an (almost) arbitrary population of chips is
placed on the vertices of a grid and does a simultaneous walk in the
Propp model, then at all times and on each vertex, the number of chips on this
vertex deviates from the expected number the random walk would have gotten
there by at most a constant. This constant is independent of the starting
configuration and the order in which each vertex serves its neighbors.
This result raises the question if all graphs do have this property. With
quite some effort, we are now able to answer this question negatively. For the
graph being an infinite -ary tree (), we show that for any
deviation there is an initial configuration of chips such that after
running the Propp model for a certain time there is a vertex with at least
more chips than expected in the random walk model. However, to achieve a
deviation of it is necessary that at least vertices
contribute by being occupied by a number of chips not divisible by at a
certain time.Comment: 15 pages, to appear in Random Structures and Algorithm
Understanding deterministic diffusion by correlated random walks
Low-dimensional periodic arrays of scatterers with a moving point particle
are ideal models for studying deterministic diffusion. For such systems the
diffusion coefficient is typically an irregular function under variation of a
control parameter. Here we propose a systematic scheme of how to approximate
deterministic diffusion coefficients of this kind in terms of correlated random
walks. We apply this approach to two simple examples which are a
one-dimensional map on the line and the periodic Lorentz gas. Starting from
suitable Green-Kubo formulas we evaluate hierarchies of approximations for
their parameter-dependent diffusion coefficients. These approximations converge
exactly yielding a straightforward interpretation of the structure of these
irregular diffusion coeficients in terms of dynamical correlations.Comment: 13 pages (revtex) with 5 figures (postscript
Transport Networks Revisited: Why Dual Graphs?
Deterministic equilibrium flows in transport networks can be investigated by
means of Markov's processes defined on the dual graph representations of the
network. Sustained movement patterns are generated by a subset of automorphisms
of the graph spanning the spatial network of a city naturally interpreted as
random walks. Random walks assign absolute scores to all nodes of a graph and
embed space syntax into Euclidean space.Comment: 12 page
Spectral Properties of Non-Unitary Band Matrices
We consider families of random non-unitary contraction operators defined as
deformations of CMV matrices which appear naturally in the study of random
quantum walks on trees or lattices. We establish several deterministic and
almost sure results about the location and nature of the spectrum of such
non-normal operators as a function of their parameters. We relate these results
to the analysis of certain random quantum walks, the dynamics of which can be
studied by means of iterates of such random non-unitary contraction operators.Comment: updated version, to appear in Annales Henri Poincar
Deterministic Approximation of Random Walks in Small Space
We give a deterministic, nearly logarithmic-space algorithm that given an undirected graph G, a positive integer r, and a set S of vertices, approximates the conductance of S in the r-step random walk on G to within a factor of 1+epsilon, where epsilon>0 is an arbitrarily small constant. More generally, our algorithm computes an epsilon-spectral approximation to the normalized Laplacian of the r-step walk.
Our algorithm combines the derandomized square graph operation [Eyal Rozenman and Salil Vadhan, 2005], which we recently used for solving Laplacian systems in nearly logarithmic space [Murtagh et al., 2017], with ideas from [Cheng et al., 2015], which gave an algorithm that is time-efficient (while ours is space-efficient) and randomized (while ours is deterministic) for the case of even r (while ours works for all r). Along the way, we provide some new results that generalize technical machinery and yield improvements over previous work. First, we obtain a nearly linear-time randomized algorithm for computing a spectral approximation to the normalized Laplacian for odd r. Second, we define and analyze a generalization of the derandomized square for irregular graphs and for sparsifying the product of two distinct graphs. As part of this generalization, we also give a strongly explicit construction of expander graphs of every size
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