1,472 research outputs found
The Niceness of Unique Sink Orientations
Random Edge is the most natural randomized pivot rule for the simplex
algorithm. Considerable progress has been made recently towards fully
understanding its behavior. Back in 2001, Welzl introduced the concepts of
\emph{reachmaps} and \emph{niceness} of Unique Sink Orientations (USO), in an
effort to better understand the behavior of Random Edge. In this paper, we
initiate the systematic study of these concepts. We settle the questions that
were asked by Welzl about the niceness of (acyclic) USO. Niceness implies
natural upper bounds for Random Edge and we provide evidence that these are
tight or almost tight in many interesting cases. Moreover, we show that Random
Edge is polynomial on at least many (possibly cyclic) USO. As
a bonus, we describe a derandomization of Random Edge which achieves the same
asymptotic upper bounds with respect to niceness and discuss some algorithmic
properties of the reachmap.Comment: An extended abstract appears in the proceedings of Approx/Random 201
Gibbs point process approximation: Total variation bounds using Stein's method
We obtain upper bounds for the total variation distance between the
distributions of two Gibbs point processes in a very general setting.
Applications are provided to various well-known processes and settings from
spatial statistics and statistical physics, including the comparison of two
Lennard-Jones processes, hard core approximation of an area interaction process
and the approximation of lattice processes by a continuous Gibbs process. Our
proof of the main results is based on Stein's method. We construct an explicit
coupling between two spatial birth-death processes to obtain Stein factors, and
employ the Georgii-Nguyen-Zessin equation for the total bound.Comment: Published in at http://dx.doi.org/10.1214/13-AOP895 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Tail estimates for homogenization theorems in random media
It is known that a random walk on among i.i.d. uniformly elliptic
random bond conductances verifies a central limit theorem. It is also known
that approximations of the covariance matrix can be obtained by considering
periodic environments. Here we estimate the speed of convergence of this
homogenization result. We obtain similar estimates for finite volume
approximations of the effective conductance and of the lowest Dirichlet
eigenvalue. A lower bound is also given for the variance of the Green function
of a random walk in a random non-negative potential.Comment: 26 page
A (2+1)-dimensional growth process with explicit stationary measures
We introduce a class of (2+1)-dimensional stochastic growth processes, that
can be seen as irreversible random dynamics of discrete interfaces.
"Irreversible" means that the interface has an average non-zero drift.
Interface configurations correspond to height functions of dimer coverings of
the infinite hexagonal or square lattice. The model can also be viewed as an
interacting driven particle system and in the totally asymmetric case the
dynamics corresponds to an infinite collection of mutually interacting
Hammersley processes.
When the dynamical asymmetry parameter equals zero, the
infinite-volume Gibbs measures (with given slope ) are
stationary and reversible. When , are not reversible any
more but, remarkably, they are still stationary. In such stationary states, we
find that the average height function at any given point grows linearly
with time with a non-zero speed: while the typical fluctuations of are
smaller than any power of as .
In the totally asymmetric case of and on the hexagonal lattice, the
dynamics coincides with the "anisotropic KPZ growth model" introduced by A.
Borodin and P. L. Ferrari. For a suitably chosen, "integrable", initial
condition (that is very far from the stationary state), they were able to
determine the hydrodynamic limit and a CLT for interface fluctuations on scale
, exploiting the fact that in that case certain space-time
height correlations can be computed exactly.Comment: 37 pages, 13 figures. v3: some references added, introduction
expanded, minor changes in the bul
The Niceness of Unique Sink Orientations
Random Edge is the most natural randomized pivot rule for the simplex algorithm. Considerable progress has been made recently towards fully understanding its behavior. Back in 2001, Welzl introduced the concepts of reachmaps and niceness of Unique Sink Orientations (USO), in an effort to better understand the behavior of Random Edge. In this paper, we initiate the systematic study of these concepts. We settle the questions that were asked by Welzl about the niceness of (acyclic) USO. Niceness implies natural upper bounds for Random Edge and we provide evidence that these are tight or almost tight in many interesting cases. Moreover, we show that Random Edge is polynomial on at least n^{Omega(2^n)} many (possibly cyclic) USO. As a bonus, we describe a derandomization of Random Edge which achieves the same asymptotic upper bounds with respect to niceness
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