3,047 research outputs found
Quantitative bounds for Markov chain convergence: Wasserstein and total variation distances
We present a framework for obtaining explicit bounds on the rate of
convergence to equilibrium of a Markov chain on a general state space, with
respect to both total variation and Wasserstein distances. For Wasserstein
bounds, our main tool is Steinsaltz's convergence theorem for locally
contractive random dynamical systems. We describe practical methods for finding
Steinsaltz's "drift functions" that prove local contractivity. We then use the
idea of "one-shot coupling" to derive criteria that give bounds for total
variation distances in terms of Wasserstein distances. Our methods are applied
to two examples: a two-component Gibbs sampler for the Normal distribution and
a random logistic dynamical system.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ238 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Monte Carlo Tests of SLE Predictions for the 2D Self-Avoiding Walk
The conjecture that the scaling limit of the two-dimensional self-avoiding
walk (SAW) in a half plane is given by the stochastic Loewner evolution (SLE)
with leads to explicit predictions about the SAW. A remarkable
feature of these predictions is that they yield not just critical exponents,
but probability distributions for certain random variables associated with the
self-avoiding walk. We test two of these predictions with Monte Carlo
simulations and find excellent agreement, thus providing numerical support to
the conjecture that the scaling limit of the SAW is SLE.Comment: TeX file using APS REVTeX 4.0. 10 pages, 5 figures (encapsulated
postscript
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