16 research outputs found
A class of stochastic games with infinitely many interacting agents related to Glauber dynamics on random graphs
We introduce and study a class of infinite-horizon nonzero-sum non-cooperative stochastic games with infinitely many interacting agents using ideas of statistical mechanics. First we show, in the general case of asymmetric interactions, the existence of a strategy that allows any player to eliminate losses after a finite random time. In the special case of symmetric interactions, we also prove that, as time goes to infinity, the game converges to a Nash equilibrium. Moreover, assuming that all agents adopt the same strategy, using arguments related to those leading to perfect simulation algorithms, spatial mixing and ergodicity are proved. In turn, ergodicity allows us to prove "fixation", i.e. that players will adopt a constant strategy after a finite time. The resulting dynamics is related to zerotemperature Glauber dynamics on random graphs of possibly infinite volume
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
Complexity Results for MCMC derived from Quantitative Bounds
This paper considers how to obtain MCMC quantitative convergence bounds which
can be translated into tight complexity bounds in high-dimensional settings. We
propose a modified drift-and-minorization approach, which establishes a
generalized drift condition defined in a subset of the state space. The subset
is called the ``large set'' and is chosen to rule out some ``bad'' states which
have poor drift property when the dimension gets large. Using the ``large set''
together with a ``centered'' drift function, a quantitative bound can be
obtained which can be translated into a tight complexity bound. As a
demonstration, we analyze a certain realistic Gibbs sampler algorithm and
obtain a complexity upper bound for the mixing time, which shows that the
number of iterations required for the Gibbs sampler to converge is constant
under certain conditions on the observed data and the initial state. It is our
hope that this modified drift-and-minorization approach can be employed in many
other specific examples to obtain complexity bounds for high-dimensional Markov
chains.Comment: 42 page
Perfect sampling from spatial mixing
We introduce a new perfect sampling technique that can be applied to general Gibbs distributions and runs in linear time if the correlation decays faster than the neighborhood growth. In particular, in graphs with subexponential neighborhood growth like [Formula: see text] , our algorithm achieves linear running time as long as Gibbs sampling is rapidly mixing. As concrete applications, we obtain the currently best perfect samplers for colorings and for monomer‐dimer models in such graphs
Linear Programming Bounds for Randomly Sampling Colorings
Here we study the problem of sampling random proper colorings of a bounded
degree graph. Let be the number of colors and let be the maximum
degree. In 1999, Vigoda showed that the Glauber dynamics is rapidly mixing for
any . It turns out that there is a natural barrier at
, below which there is no one-step coupling that is contractive,
even for the flip dynamics.
We use linear programming and duality arguments to guide our construction of
a better coupling. We fully characterize the obstructions to going beyond
. These examples turn out to be quite brittle, and even starting
from one, they are likely to break apart before the flip dynamics changes the
distance between two neighboring colorings. We use this intuition to design a
variable length coupling that shows that the Glauber dynamics is rapidly mixing
for any where . This is the first improvement to Vigoda's analysis that
holds for general graphs.Comment: 30 pages, 3 figures; fixed some typo