12,676 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
Critical Percolation Exploration Path and SLE(6): a Proof of Convergence
It was argued by Schramm and Smirnov that the critical site percolation
exploration path on the triangular lattice converges in distribution to the
trace of chordal SLE(6). We provide here a detailed proof, which relies on
Smirnov's theorem that crossing probabilities have a conformally invariant
scaling limit (given by Cardy's formula). The version of convergence to SLE(6)
that we prove suffices for the Smirnov-Werner derivation of certain critical
percolation crossing exponents and for our analysis of the critical percolation
full scaling limit as a process of continuum nonsimple loops.Comment: 45 pages, 14 figures; revised version following the comments of a
refere
A lagrangian approach to weakly coupled Hamilton-Jacobi systems
We study a class of weakly coupled Hamilton–Jacobi systems with a specific
aim to perform a qualitative analysis in the spirit of weak KAM theory. Our main
achievement is the definition of a family of related action functionals containing the
Lagrangians obtained by duality from the Hamiltonians of the system. We use them to
characterize, by means of a suitable estimate, all the subsolutions of the system, and
to explicitly represent some subsolutions enjoying an additional maximality property. A
crucial step for our analysis is to put the problem in a suitable random frame. Only
some basic knowledge of measure theory is required, and the presentation is accessible
to readers without background in probability
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
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