807 research outputs found
Mod-phi convergence I: Normality zones and precise deviations
In this paper, we use the framework of mod- convergence to prove
precise large or moderate deviations for quite general sequences of real valued
random variables , which can be lattice or
non-lattice distributed. We establish precise estimates of the fluctuations
, instead of the usual estimates for the rate of
exponential decay . Our approach provides us with a
systematic way to characterise the normality zone, that is the zone in which
the Gaussian approximation for the tails is still valid. Besides, the residue
function measures the extent to which this approximation fails to hold at the
edge of the normality zone.
The first sections of the article are devoted to a proof of these abstract
results and comparisons with existing results. We then propose new examples
covered by this theory and coming from various areas of mathematics: classical
probability theory, number theory (statistics of additive arithmetic
functions), combinatorics (statistics of random permutations), random matrix
theory (characteristic polynomials of random matrices in compact Lie groups),
graph theory (number of subgraphs in a random Erd\H{o}s-R\'enyi graph), and
non-commutative probability theory (asymptotics of random character values of
symmetric groups). In particular, we complete our theory of precise deviations
by a concrete method of cumulants and dependency graphs, which applies to many
examples of sums of "weakly dependent" random variables. The large number as
well as the variety of examples hint at a universality class for second order
fluctuations.Comment: 103 pages. New (final) version: multiple small improvements ; a new
section on mod-Gaussian convergence coming from the factorization of the
generating function ; the multi-dimensional results have been moved to a
forthcoming paper ; and the introduction has been reworke
Asymptotic of geometrical navigation on a random set of points of the plane
A navigation on a set of points is a rule for choosing which point to
move to from the present point in order to progress toward a specified target.
We study some navigations in the plane where is a non uniform Poisson point
process (in a finite domain) with intensity going to . We show the
convergence of the traveller path lengths, the number of stages done, and the
geometry of the traveller trajectories, uniformly for all starting points and
targets, for several navigations of geometric nature. Other costs are also
considered. This leads to asymptotic results on the stretch factors of random
Yao-graphs and random -graphs
Counting points on varieties over finite fields related to a conjecture of Kontsevich
We describe a characteristic-free algorithm for âreducingâ an algebraic variety defined by the vanishing of a set of integer polynomials. In very special cases, the algorithm can be used to decide whether the number of points on a variety, as the ground field varies over finite fields, is a polynomial function of the size of the field. The algorithm is then used to investigate a conjecture of Kontsevich regarding the number of points on a variety associated with the set of spanning trees of any graph. We also prove several theorems describing properties of a (hypothetical) minimal counterexample to the conjecture, and produce counterexamples to some related conjectures.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43265/1/26_2005_Article_BF01608531.pd
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