10,200 research outputs found
Rapid mixing of Swendsen-Wang dynamics in two dimensions
We prove comparison results for the Swendsen-Wang (SW) dynamics, the
heat-bath (HB) dynamics for the Potts model and the single-bond (SB) dynamics
for the random-cluster model on arbitrary graphs. In particular, we prove that
rapid mixing of HB implies rapid mixing of SW on graphs with bounded maximum
degree and that rapid mixing of SW and rapid mixing of SB are equivalent.
Additionally, the spectral gap of SW and SB on planar graphs is bounded from
above and from below by the spectral gap of these dynamics on the corresponding
dual graph with suitably changed temperature.
As a consequence we obtain rapid mixing of the Swendsen-Wang dynamics for the
Potts model on the two-dimensional square lattice at all non-critical
temperatures as well as rapid mixing for the two-dimensional Ising model at all
temperatures. Furthermore, we obtain new results for general graphs at high or
low enough temperatures.Comment: Ph.D. thesis, 66 page
Robustness of large-scale stochastic matrices to localized perturbations
Upper bounds are derived on the total variation distance between the
invariant distributions of two stochastic matrices differing on a subset W of
rows. Such bounds depend on three parameters: the mixing time and the minimal
expected hitting time on W for the Markov chain associated to one of the
matrices; and the escape time from W for the Markov chain associated to the
other matrix. These results, obtained through coupling techniques, prove
particularly useful in scenarios where W is a small subset of the state space,
even if the difference between the two matrices is not small in any norm.
Several applications to large-scale network problems are discussed, including
robustness of Google's PageRank algorithm, distributed averaging and consensus
algorithms, and interacting particle systems.Comment: 12 pages, 4 figure
Comparison of Swendsen-Wang and Heat-Bath Dynamics
We prove that the spectral gap of the Swendsen-Wang process for the Potts
model on graphs with bounded degree is bounded from below by some constant
times the spectral gap of any single-spin dynamics. This implies rapid mixing
of the Swendsen-Wang process for the two-dimensional Potts model at all
temperatures above the critical one, as well as rapid mixing at the critical
temperature for the Ising model. After this we introduce a modified version of
the Swendsen-Wang algorithm for planar graphs and prove rapid mixing for the
two-dimensional Potts models at all non-critical temperatures.Comment: 22 pages, 1 figur
Quantum logarithmic Sobolev inequalities and rapid mixing
A family of logarithmic Sobolev inequalities on finite dimensional quantum
state spaces is introduced. The framework of non-commutative \bL_p-spaces is
reviewed and the relationship between quantum logarithmic Sobolev inequalities
and the hypercontractivity of quantum semigroups is discussed. This
relationship is central for the derivation of lower bounds for the logarithmic
Sobolev (LS) constants. Essential results for the family of inequalities are
proved, and we show an upper bound to the generalized LS constant in terms of
the spectral gap of the generator of the semigroup. These inequalities provide
a framework for the derivation of improved bounds on the convergence time of
quantum dynamical semigroups, when the LS constant and the spectral gap are of
the same order. Convergence bounds on finite dimensional state spaces are
particularly relevant for the field of quantum information theory. We provide a
number of examples, where improved bounds on the mixing time of several
semigroups are obtained; including the depolarizing semigroup and quantum
expanders.Comment: Updated manuscript, 30 pages, no figure
Rapid mixing of Swendsen-Wang and single-bond dynamics in two dimensions
We prove that the spectral gap of the Swendsen-Wang dynamics for the
random-cluster model on arbitrary graphs with m edges is bounded above by 16 m
log m times the spectral gap of the single-bond (or heat-bath) dynamics. This
and the corresponding lower bound imply that rapid mixing of these two dynamics
is equivalent.
Using the known lower bound on the spectral gap of the Swendsen-Wang dynamics
for the two dimensional square lattice of side length L at high
temperatures and a result for the single-bond dynamics on dual graphs, we
obtain rapid mixing of both dynamics on at all non-critical
temperatures. In particular this implies, as far as we know, the first proof of
rapid mixing of a classical Markov chain for the Ising model on at all
temperatures.Comment: 20 page
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