44,082 research outputs found
Metric Construction, Stopping Times and Path Coupling
In this paper we examine the importance of the choice of metric in path
coupling, and the relationship of this to \emph{stopping time analysis}. We
give strong evidence that stopping time analysis is no more powerful than
standard path coupling. In particular, we prove a stronger theorem for path
coupling with stopping times, using a metric which allows us to restrict
analysis to standard one-step path coupling. This approach provides insight for
the design of non-standard metrics giving improvements in the analysis of
specific problems.
We give illustrative applications to hypergraph independent sets and SAT
instances, hypergraph colourings and colourings of bipartite graphs.Comment: 21 pages, revised version includes statement and proof of general
stopping times theorem (section 2.2), and additonal remarks in section
Path Coupling Using Stopping Times and Counting Independent Sets and Colourings in Hypergraphs
We give a new method for analysing the mixing time of a Markov chain using
path coupling with stopping times. We apply this approach to two hypergraph
problems. We show that the Glauber dynamics for independent sets in a
hypergraph mixes rapidly as long as the maximum degree Delta of a vertex and
the minimum size m of an edge satisfy m>= 2Delta+1. We also show that the
Glauber dynamics for proper q-colourings of a hypergraph mixes rapidly if m>= 4
and q > Delta, and if m=3 and q>=1.65Delta. We give related results on the
hardness of exact and approximate counting for both problems.Comment: Simpler proof of main theorem. Improved bound on mixing time. 19
page
Particle Stirring in Turbulent Gas Disks: Including Orbital Oscillations
We describe the diffusion and random velocities of solid particles due to
stochastic forcing by turbulent gas. We include the orbital dynamics of
Keplerian disks, both in-plane epicycles and vertical oscillations. We obtain a
new result for the diffusion of solids. The Schmidt number (ratio of gas to
particle diffusivity) is Sc = 1 + (Omega t_stop)^2, in terms of the particle
stopping time, t_stop, and the orbital frequency, Omega. The standard result,
Sc = 1 + t_stop/t_eddy, in terms of the eddy turnover time, t_eddy, is shown to
be incorrect. The main difference is that Sc rises quadratically, not linearly,
with stopping time. Consequently, particles larger than ~ 10 cm in
protoplanetary disks will suffer less radial diffusion and will settle closer
to the midplane. Such a layer of boulders would be more prone to gravitational
collapse. Our predictions of RMS speeds, vertical scale height and diffusion
coefficients will help interpret numerical simulations. We confirm previous
results for the vertical stirring of particles (scale heights and random
velocities), and add a correction for arbitrary ratios of eddy to orbital
times. The particle layer becomes thinner for t_eddy > 1/Omega, with the
strength of turbulent diffusion held fixed. We use two analytic techniques --
the Hinze-Tchen formalism and the Fokker-Planck equation with velocity
diffusion -- with identical results when the regimes of validity overlap. We
include simple physical arguments for the scaling of our results.Comment: 17 pages, 7 figures, 2 tables, accepted to Icaru
Long time behavior of telegraph processes under convex potentials
We study the long-time behavior of variants of the telegraph process with
position-dependent jump-rates, which result in a monotone gradient-like drift
toward the origin. We compute their invariant laws and obtain, via
probabilistic couplings arguments, some quantitative estimates of the total
variation distance to equilibrium. Our techniques extend ideas previously
developed for a simplified piecewise deterministic Markov model of bacterial
chemotaxis.Comment: 26 pages, 3 figure
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