3,005 research outputs found
Functions of random walks on hyperplane arrangements
Many seemingly disparate Markov chains are unified when viewed as random
walks on the set of chambers of a hyperplane arrangement. These include the
Tsetlin library of theoretical computer science and various shuffling schemes.
If only selected features of the chains are of interest, then the mixing times
may change. We study the behavior of hyperplane walks, viewed on a
subarrangement of a hyperplane arrangement. These include many new examples,
for instance a random walk on the set of acyclic orientations of a graph. All
such walks can be treated in a uniform fashion, yielding diagonalizable
matrices with known eigenvalues, stationary distribution and good rates of
convergence to stationarity.Comment: Final version; Section 4 has been split into two section
Deterministic Walks in Quenched Random Environments of Chaotic Maps
This paper concerns the propagation of particles through a quenched random
medium. In the one- and two-dimensional models considered, the local dynamics
is given by expanding circle maps and hyperbolic toral automorphisms,
respectively. The particle motion in both models is chaotic and found to
fluctuate about a linear drift. In the proper scaling limit, the cumulative
distribution function of the fluctuations converges to a Gaussian one with
system dependent variance while the density function shows no convergence to
any function. We have verified our analytical results using extreme precision
numerical computations.Comment: 18 pages, 9 figure
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