21,353 research outputs found
Matrix norms and rapid mixing for spin systems
We give a systematic development of the application of matrix norms to rapid
mixing in spin systems. We show that rapid mixing of both random update Glauber
dynamics and systematic scan Glauber dynamics occurs if any matrix norm of the
associated dependency matrix is less than 1. We give improved analysis for the
case in which the diagonal of the dependency matrix is (as in heat
bath dynamics). We apply the matrix norm methods to random update and
systematic scan Glauber dynamics for coloring various classes of graphs. We
give a general method for estimating a norm of a symmetric nonregular matrix.
This leads to improved mixing times for any class of graphs which is hereditary
and sufficiently sparse including several classes of degree-bounded graphs such
as nonregular graphs, trees, planar graphs and graphs with given tree-width and
genus.Comment: Published in at http://dx.doi.org/10.1214/08-AAP532 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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
Universality of cutoff for the Ising model
On any locally-finite geometry, the stochastic Ising model is known to be
contractive when the inverse-temperature is small enough, via classical
results of Dobrushin and of Holley in the 1970's. By a general principle
proposed by Peres, the dynamics is then expected to exhibit cutoff. However, so
far cutoff for the Ising model has been confirmed mainly for lattices, heavily
relying on amenability and log Sobolev inequalities. Without these, cutoff was
unknown at any fixed , no matter how small, even in basic examples
such as the Ising model on a binary tree or a random regular graph.
We use the new framework of information percolation to show that, in any
geometry, there is cutoff for the Ising model at high enough temperatures.
Precisely, on any sequence of graphs with maximum degree , the Ising model
has cutoff provided that for some absolute constant
(a result which, up to the value of , is best possible). Moreover, the
cutoff location is established as the time at which the sum of squared
magnetizations drops to 1, and the cutoff window is , just as when
.
Finally, the mixing time from almost every initial state is not more than a
factor of faster then the worst one (with
as ), whereas the uniform starting state is at
least times faster.Comment: 26 pages, 2 figures. Companion paper to arXiv:1401.606
Phase transition for the mixing time of the Glauber dynamics for coloring regular trees
We prove that the mixing time of the Glauber dynamics for random k-colorings
of the complete tree with branching factor b undergoes a phase transition at
. Our main result shows nearly sharp bounds on the mixing
time of the dynamics on the complete tree with n vertices for
colors with constant C. For we prove the mixing time is
. On the other side, for the mixing time
experiences a slowing down; in particular, we prove it is
and . The critical point C=1
is interesting since it coincides (at least up to first order) with the
so-called reconstruction threshold which was recently established by Sly. The
reconstruction threshold has been of considerable interest recently since it
appears to have close connections to the efficiency of certain local
algorithms, and this work was inspired by our attempt to understand these
connections in this particular setting.Comment: Published in at http://dx.doi.org/10.1214/11-AAP833 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Glauber dynamics on trees:Boundary conditions and mixing time
We give the first comprehensive analysis of the effect of boundary conditions
on the mixing time of the Glauber dynamics in the so-called Bethe
approximation. Specifically, we show that spectral gap and the log-Sobolev
constant of the Glauber dynamics for the Ising model on an n-vertex regular
tree with plus-boundary are bounded below by a constant independent of n at all
temperatures and all external fields. This implies that the mixing time is
O(log n) (in contrast to the free boundary case, where it is not bounded by any
fixed polynomial at low temperatures). In addition, our methods yield simpler
proofs and stronger results for the spectral gap and log-Sobolev constant in
the regime where there are multiple phases but the mixing time is insensitive
to the boundary condition. Our techniques also apply to a much wider class of
models, including those with hard-core constraints like the antiferromagnetic
Potts model at zero temperature (proper colorings) and the hard--core lattice
gas (independent sets)
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
Mixing times of lozenge tiling and card shuffling Markov chains
We show how to combine Fourier analysis with coupling arguments to bound the
mixing times of a variety of Markov chains. The mixing time is the number of
steps a Markov chain takes to approach its equilibrium distribution. One
application is to a class of Markov chains introduced by Luby, Randall, and
Sinclair to generate random tilings of regions by lozenges. For an L X L region
we bound the mixing time by O(L^4 log L), which improves on the previous bound
of O(L^7), and we show the new bound to be essentially tight. In another
application we resolve a few questions raised by Diaconis and Saloff-Coste, by
lower bounding the mixing time of various card-shuffling Markov chains. Our
lower bounds are within a constant factor of their upper bounds. When we use
our methods to modify a path-coupling analysis of Bubley and Dyer, we obtain an
O(n^3 log n) upper bound on the mixing time of the Karzanov-Khachiyan Markov
chain for linear extensions.Comment: 39 pages, 8 figure
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