21,579 research outputs found
Zeros of random tropical polynomials, random polytopes and stick-breaking
For , let be independent and identically
distributed random variables with distribution with support .
The number of zeros of the random tropical polynomials is also the number of faces of the lower convex
hull of the random points in . We show that this
number, , satisfies a central limit theorem when has polynomial decay
near . Specifically, if near behaves like a
distribution for some , then has the same asymptotics as the
number of renewals on the interval of a renewal process with
inter-arrival distribution . Our proof draws on connections
between random partitions, renewal theory and random polytopes. In particular,
we obtain generalizations and simple proofs of the central limit theorem for
the number of vertices of the convex hull of uniform random points in a
square. Our work leads to many open problems in stochastic tropical geometry,
the study of functionals and intersections of random tropical varieties.Comment: 22 pages, 5 figure
An Update on Local Universality Limits for Correlation Functions Generated by Unitary Ensembles
We survey the current status of universality limits for -point correlation
functions in the bulk and at the edge for unitary ensembles, primarily when the
limiting kernels are Airy, Bessel, or Sine kernels. In particular, we consider
underlying measures on compact intervals, and fixed and varying exponential
weights, as well as universality limits for a variety of orthogonal systems.
The scope of the survey is quite narrow: we do not consider ensembles
for , nor general Hermitian matrices with independent entries,
let alone more general settings. We include some open problems
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