116 research outputs found
Equality of averaged and quenched large deviations for random walks in random environments in dimensions four and higher
We consider large deviations for nearest-neighbor random walk in a uniformly
elliptic i.i.d. environment. It is easy to see that the quenched and the
averaged rate functions are not identically equal. When the dimension is at
least four and Sznitman's transience condition (T) is satisfied, we prove that
these rate functions are finite and equal on a closed set whose interior
contains every nonzero velocity at which the rate functions vanish.Comment: 17 pages. Minor revision. In particular, note the change in the title
of the paper. To appear in Probability Theory and Related Fields
Selected recollections of my relationship with Leo Breiman
During the period 1962--1964, I had a tenure track Assistant Professorship in
Mathematics at Cornell University in Ithaca, New York, where I did research in
probability theory, especially on linear diffusion processes. Being somewhat
lonely there and not liking the cold winter weather, I decided around the
beginning of 1964 to try to get a job in the Mathematics Department at UCLA, in
the city in which I was born and raised. At that time, Leo Breiman was an
Associate Professor in that department. Presumably, he liked my research on
linear diffusion processes and other research as well, since the department
offered me a tenure track Assistant Professorship, which I happily accepted.
During the Summer of 1965, I worked on various projects with Sidney Port, then
at RAND Corporation, especially on random walks and related material. I was
promoted to Associate Professor, effective in Fall, 1966, presumably thanks in
part to Leo. Early in 1966, I~was surprised to be asked by Leo to participate
in a department meeting called to discuss the possible hiring of Sidney. The
conclusion was that Sidney was hired as Associate Professor in the department,
as of Fall, 1966. Leo communicated to me his view that he thought that Sidney
and I worked well together, which is why he had urged the department to hire
Sidney. Anyhow, Sidney and I had a very fruitful and enjoyable collaboration in
probability and, to a much lesser extent, in theoretical statistics, for a
number of years thereafter.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS431 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Quenched large deviations for multidimensional random walk in random environment with holding times
We consider a random walk in random environment with random holding times,
that is, the random walk jumping to one of its nearest neighbors with some
transition probability after a random holding time. Both the transition
probabilities and the laws of the holding times are randomly distributed over
the integer lattice. Our main result is a quenched large deviation principle
for the position of the random walk. The rate function is given by the Legendre
transform of the so-called Lyapunov exponents for the Laplace transform of the
first passage time. By using this representation, we derive some asymptotics of
the rate function in some special cases.Comment: This is the corrected version of the paper. 24 page
Variational formulas and cocycle solutions for directed polymer and percolation models
We discuss variational formulas for the law of large numbers limits of certain models of motion in a random medium: namely, the limiting time constant for last-passage percolation and the limiting free energy for directed polymers. The results are valid for models in arbitrary dimension, steps of the admissible paths can be general, the environment process is ergodic under spatial translations, and the potential accumulated along a path can depend on the environment and the next step of the path. The variational formulas come in two types: one minimizes over gradient-like cocycles, and another one maximizes over invariant measures on the space of environments and paths. Minimizing cocycles can be obtained from Busemann functions when these can be proved to exist. The results are illustrated through 1+1 dimensional exactly solvable examples, periodic examples, and polymers in weak disorder
Cut Points and Diffusions in Random Environment
In this article we investigate the asymptotic behavior of a new class of
multi-dimensional diffusions in random environment. We introduce cut times in
the spirit of the work done by Bolthausen, Sznitman and Zeitouni, see [4], in
the discrete setting providing a decoupling effect in the process. This allows
us to take advantage of an ergodic structure to derive a strong law of large
numbers with possibly vanishing limiting velocity and a central limit theorem
under the quenched measure.Comment: 44 pages; accepted for publication in "Journal of Theoretical
Probability
Random walks in random Dirichlet environment are transient in dimension
We consider random walks in random Dirichlet environment (RWDE) which is a
special type of random walks in random environment where the exit probabilities
at each site are i.i.d. Dirichlet random variables. On , RWDE are
parameterized by a -uplet of positive reals. We prove that for all values
of the parameters, RWDE are transient in dimension . We also prove that
the Green function has some finite moments and we characterize the finite
moments. Our result is more general and applies for example to finitely
generated symmetric transient Cayley graphs. In terms of reinforced random
walks it implies that directed edge reinforced random walks are transient for
.Comment: New version published at PTRF with an analytic proof of lemma
Level 1 quenched large deviation principle for random walk in dynamic random environment
Abstract Consider a random walk in a time-dependent random environment on the lattice Z d . Recently, Rassoul-Agha, SeppÀlÀinen and Yilmaz [RSY11] proved a general large deviation principle under mild ergodicity assumptions on the random environment for such a random walk, establishing first level 2 and 3 large deviation principles. Here we present two alternative short proofs of the level 1 large deviations under mild ergodicity assumptions on the environment: one for the continuous time case and another one for the discrete time case. Both proofs provide the existence, continuity and convexity of the rate function. Our methods are based on the use of the sub-additive ergodic theorem as presented by Varadhan in [V03]
The rarity of terrestrial gamma-ray flashes
We report on the first search for Terrestrial Gamma-ray Flashes (TGFs) from altitudes where they are thought to be produced. The Airborne Detector for Energetic Lightning Emissions (ADELE), an array of gamma-ray detectors, was flown near the tops of Florida thunderstorms in August/September 2009. The plane passed within 10 km horizontal distance of 1213 lightning discharges and only once detected a TGF. If these discharges had produced TGFs of the same intensity as those seen from space, every one should have been seen by ADELE. Separate and significant nondetections are established for intracloud lightning, negative cloud-to-ground lightning, and narrow bipolar events. We conclude that TGFs are not a primary triggering mechanism for lightning. We estimate the TGF-to-flash ratio to be on the order of 10^(â2) to 10^(â3) and show that TGF intensities cannot follow the well-known power-law distribution seen in earthquakes and solar flares, due to our limits on the presence of faint events
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