11 research outputs found
On the mixing time and spectral gap for birth and death chains
For birth and death chains, we derive bounds on the spectral gap and mixing
time in terms of birth and death rates. Together with the results of Ding et
al. in 2010, this provides a criterion for the existence of a cutoff in terms
of the birth and death rates. A variety of illustrative examples are treated
The power of averaging at two consecutive time steps: Proof of a mixing conjecture by Aldous and Fill
Let be an irreducible reversible discrete time
Markov chain on a finite state space . Denote its transition matrix by
. To avoid periodicity issues (and thus ensuring convergence to equilibrium)
one often considers the continuous-time version of the chain
whose kernel is given by . Another possibility is to consider the associated averaged chain
, whose distribution at time is
obtained by replacing by .
A sequence of Markov chains is said to exhibit (total-variation) cutoff if
the convergence to stationarity in total-variation distance is abrupt. Let
be a sequence of irreducible reversible
discrete time Markov chains. In this work we prove that the sequence of
associated continuous-time chains exhibits total-variation cutoff around time
iff the sequence of the associated averaged chains exhibits
total-variation cutoff around time . Moreover, we show that the width of
the cutoff window for the sequence of associated averaged chains is at most
that of the sequence of associated continuous-time chains. In fact, we
establish more precise quantitative relations between the mixing-times of the
continuous-time and the averaged versions of a reversible Markov chain, which
provide an affirmative answer to a problem raised by Aldous and Fill.Non
Cutoff for the noisy voter model
Given a continuous time Markov Chain on a finite set , the
associated noisy voter model is the continuous time Markov chain on
, which evolves in the following way: (1) for each two sites and
in , the state at site changes to the value of the state at site
at rate ; (2) each site rerandomizes its state at rate 1. We show that
if there is a uniform bound on the rates and the corresponding
stationary distributions are almost uniform, then the mixing time has a sharp
cutoff at time with a window of order 1. Lubetzky and Sly proved
cutoff with a window of order 1 for the stochastic Ising model on toroids; we
obtain the special case of their result for the cycle as a consequence of our
result. Finally, we consider the model on a star and demonstrate the surprising
phenomenon that the time it takes for the chain started at all ones to become
close in total variation to the chain started at all zeros is of smaller order
than the mixing time.Comment: Published at http://dx.doi.org/10.1214/15-AAP1108 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Sensitivity of mixing times of Cayley graphs
We show that the total variation mixing time is neither quasi-isometry
invariant nor robust, even for Cayley graphs. The Cayley graphs serving as an
example have unbounded degrees. For non-transitive graphs we show bounded
degree graphs for which the mixing time from the worst point for one graph is
asymptotically smaller than the mixing time from the best point of the random
walk on a network obtained by increasing some of the edge weights from 1 to
.Comment: 28 pages, 1 figur
MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications
Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described