Algebraic Geometry Seminar
Steve Zelditch
Northwestern
Probability that a random configuration of N points is the zero set of a holomorphic section
Abstract: Let $X$ be a Riemann surface and let $E_N \to Pic^N$ be the vector bundle
consisting of pairs $(L, s)$ of a line bundle of degree $N$ and a holomorphic section
of
$L$. As a generalization of ``random polynomial" (the case $X = CP^1$), we put a
Gaussian
type probability measure on $E_N$, i.e. we choose $L$ at random and then s at random
from $H^0(X, L)$. We use a Hermitian metric $h_L$ on $L$ and a measure $\nu$ on $X$ to
define
the Gaussian measure on
$E_N$. This probability measure induces a probability measure on the configuration
space $S^N X$ of $N$ points, and we can ask, what is the probability that a given
configuration of $N$ points is the zero set of a random section? The bosonization
formulae of string theory are used to determine the induced
measure on configurations of points. It turns out that as $N \to \infty$, the
configurations concentrate very quickly on a certain equilibrium configuration
determined by $(h, L)$.
No prior knowledge of probability theory (or bosonization formulae) is assumed.
This
result is a generalization of joint work with O. Zeitouni
in the genus zero case and with B. Shiffman on the planar case.
Wednesday March 16, 2011 at 4:00 PM in SEO 1227