Large Deviations and Effective Equidistribution

Effective Equidistribution and Spectral Gap

Abstract. In these notes we discuss some equidistribution problems with the aim to give reasonable error rates, i.e. we are interested in effective statements. We motivate some arguments by studying a concrete problem on a two-torus, and then describe recent results on the equidistribution of semisimple orbits obtained in joint work with G. Margulis and A. Venkatesh. We end by studying the rela...

Large Deviations

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. 1. Large deviations for sums. The role of " large deviations " is best understood through an example. Suppose that X random variables, for instance, normally distributed with mean zero and variance 1. Then, E e θ(X 1 +···+X n) = E[e θX ...

Effective equidistribution of eigenvalues of Hecke operators

Article history: Received 13 October 2007 Revised 9 May 2008 Available online 17 December 2008 Communicated by David Goss In 1997, Serre proved an equidistribution theorem for eigenvalues of Hecke operators on the space S(N,k) of cusp forms of weight k and level N . In this paper, we derive an effective version of Serre’s theorem. As a consequence, we estimate, for a given d and prime p coprime...

Large Deviations for Large Capacity

In this paper, we study large deviations of large capacity loss networks with xed routing. We use two-level modelling for the loss networks: the call level and the cell level. At the call level, a call request is accepted if it succeeds an admission test. The test is based on a polyhedral set of the number of calls in progress when a new call arrives. After being accepted, a call then transmits...