513 research outputs found
Special invited paper. Large deviations
This paper is based on Wald Lectures given at the annual meeting of the IMS
in Minneapolis during August 2005. It is a survey of the theory of large
deviations.Comment: Published in at http://dx.doi.org/10.1214/07-AOP348 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Large deviations for the Yang-Mills measure on a compact surface
We prove the first mathematical result relating the Yang-Mills measure on a
compact surface and the Yang-Mills energy. We show that, at the small volume
limit, the Yang-Mills measures satisfy a large deviation principle with a rate
function which is expressed in a simple and natural way in terms of the
Yang-Mills energy
The large deviation principle for the Erd\H{o}s-R\'enyi random graph
What does an Erdos-Renyi graph look like when a rare event happens? This
paper answers this question when p is fixed and n tends to infinity by
establishing a large deviation principle under an appropriate topology. The
formulation and proof of the main result uses the recent development of the
theory of graph limits by Lovasz and coauthors and Szemeredi's regularity lemma
from graph theory. As a basic application of the general principle, we work out
large deviations for the number of triangles in G(n,p). Surprisingly, even this
simple example yields an interesting double phase transition.Comment: 24 pages. To appear in European J. Comb. (special issue on graph
limits
Brownian Occupation Measures, Compactness and Large Deviations
In proving large deviation estimates, the lower bound for open sets and upper
bound for compact sets are essentially local estimates. On the other hand, the
upper bound for closed sets is global and compactness of space or an
exponential tightness estimate is needed to establish it. In dealing with the
occupation measure L_t(A)=\frac{1}{t}\int_0^t{\1}_A(W_s) \d s of the
dimensional Brownian motion, which is not positive recurrent, there is no
possibility of exponential tightness. The space of probability distributions
can be compactified by replacing the usual topology of
weak convergence by the vague toplogy, where the space is treated as the dual
of continuous functions with compact support. This is essentially the one point
compactification of by adding a point at that results in the
compactification of by allowing some mass to escape to the
point at . If one were to use only test functions that are continuous
and vanish at then the compactification results in the space of
sub-probability distributions by ignoring the mass
at .
The main drawback of this compactification is that it ignores the underlying
translation invariance. More explicitly, we may be interested in the space of
equivalence classes of orbits under the action of the translation group on . There are problems for which it is natural to compactify this space
of orbits. We will provide such a compactification, prove a large deviation
principle there and give an application to a relevant problem.Comment: Minor revision. To appear in the "Annals of Probability
Large deviations for random walk in a random environment
In this work, we study the large deviation properties of random walk in a
random environment on with .
We start with the quenched case, take the point of view of the particle, and
prove the large deviation principle (LDP) for the pair empirical measure of the
environment Markov chain. By an appropriate contraction, we deduce the quenched
LDP for the mean velocity of the particle and obtain a variational formula for
the corresponding rate function . We propose an Ansatz for the minimizer
of this formula. This Ansatz is easily verified when .
In his 2003 paper, Varadhan proves the averaged LDP for the mean velocity and
gives a variational formula for the corresponding rate function . Under
the non-nestling assumption (resp. Kalikow's condition), we show that is
strictly convex and analytic on a non-empty open set , and that
the true velocity is an element (resp. in the closure) of
. We then identify the minimizer of Varadhan's variational formula
at any .
For walks in high dimension, we believe that and agree on a set
with non-empty interior. We prove this for space-time walks when the dimension
is at least 3+1. In the latter case, we show that the cheapest way to condition
the asymptotic mean velocity of the particle to be equal to any close to
is to tilt the transition kernel of the environment Markov chain via a
Doob -transform.Comment: 82 pages. PhD thesis. Advisor: S.R.S. Varadha
Low Cost Swarm Based Diligent Cargo Transit System
The goal of this paper is to present the design and development of a low cost
cargo transit system which can be adapted in developing countries like India
where there is abundant and cheap human labour which makes the process of
automation in any industry a challenge to innovators. The need of the hour is
an automation system that can diligently transfer cargo from one place to
another and minimize human intervention in the cargo transit industry.
Therefore, a solution is being proposed which could effectively bring down
human labour and the resources needed to implement them. The reduction in human
labour and resources is achieved by the use of low cost components and very
limited modification of the surroundings and the existing vehicles themselves.
The operation of the cargo transit system has been verified and the relevant
results are presented. An economical and robust cargo transit system is
designed and implemented.Comment: 6 pages, 9 figures, 1 block diagra
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