416 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
Optimal strong stationary times for random walks on the chambers of a hyperplane arrangement
This paper studies Markov chains on the chambers of real hyperplane
arrangements, a model that generalizes famous examples, such as the Tsetlin
library and riffle shuffles. We discuss cutoff for the Tsetlin library for
general weights, and we give an exact formula for the separation distance for
the hyperplane arrangement walk. We introduce lower bounds, which allow for the
first time to study cutoff for hyperplane arrangement walks under certain
conditions. Using similar techniques, we also prove a uniform lower bound for
the mixing time of Glauber dynamics on a monotone system.Comment: 13 pages. arXiv admin note: text overlap with arXiv:1605.0833
Poset topology and homological invariants of algebras arising in algebraic combinatorics
We present a beautiful interplay between combinatorial topology and
homological algebra for a class of monoids that arise naturally in algebraic
combinatorics. We explore several applications of this interplay. For instance,
we provide a new interpretation of the Leray number of a clique complex in
terms of non-commutative algebra.
R\'esum\'e. Nous pr\'esentons une magnifique interaction entre la topologie
combinatoire et l'alg\`ebre homologique d'une classe de mono\"ides qui figurent
naturellement dans la combinatoire alg\'ebrique. Nous explorons plusieurs
applications de cette interaction. Par exemple, nous introduisons une nouvelle
interpr\'etation du nombre de Leray d'un complexe de clique en termes de la
dimension globale d'une certaine alg\`ebre non commutative.Comment: This is an extended abstract surveying the results of arXiv:1205.1159
and an article in preparation. 12 pages, 3 Figure
Semigroups, rings, and Markov chains
We analyze random walks on a class of semigroups called ``left-regular
bands''. These walks include the hyperplane chamber walks of Bidigare, Hanlon,
and Rockmore. Using methods of ring theory, we show that the transition
matrices are diagonalizable and we calculate the eigenvalues and
multiplicities. The methods lead to explicit formulas for the projections onto
the eigenspaces. As examples of these semigroup walks, we construct a random
walk on the maximal chains of any distributive lattice, as well as two random
walks associated with any matroid. The examples include a q-analogue of the
Tsetlin library. The multiplicities of the eigenvalues in the matroid walks are
``generalized derangement numbers'', which may be of independent interest.Comment: To appear in J. Theoret. Proba
Riffle shuffles with biased cuts
The well-known Gilbert-Shannon-Reeds model for riffle shuffles assumes that
the cards are initially cut 'about in half' and then riffled together. We
analyze a natural variant where the initial cut is biased. Extending results of
Fulman (1998), we show a sharp cutoff in separation and L-infinity distances.
This analysis is possible due to the close connection between shuffling and
quasisymmetric functions along with some complex analysis of a generating
function.Comment: 10 page
The shape of a random affine Weyl group element and random core partitions
Let be a finite Weyl group and be the corresponding affine
Weyl group. We show that a large element in , randomly generated by
(reduced) multiplication by simple generators, almost surely has one of
-specific shapes. Equivalently, a reduced random walk in the regions of
the affine Coxeter arrangement asymptotically approaches one of -many
directions. The coordinates of this direction, together with the probabilities
of each direction can be calculated via a Markov chain on . Our results,
applied to type , show that a large random -core obtained
from the natural growth process has a limiting shape which is a
piecewise-linear graph. In this case, our random process is a periodic analogue
of TASEP, and our limiting shapes can be compared with Rost's theorem on the
limiting shape of TASEP.Comment: Published at http://dx.doi.org/10.1214/14-AOP915 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Eigenvectors for a random walk on a hyperplane arrangement
We find explicit eigenvectors for the transition matrix of a random walk due
to Bidegare, Hanlon and Rockmore. This is accomplished by using Brown and
Diaconis' analysis of its stationary distribution, together with some
combinatorics of functions on the face lattice of a hyperplane arrangement, due
to Gelfand and Varchenko.Comment: 13 pages; to appear in Advances in Applied Mathematic
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