arXiv subjects

Scott Zrebiec

Publications and source records attributed to Scott Zrebiec.

5 recordsLinked to original sources

arXiv2008

Estimates on the Probability of Outliers for Real Random Bargmann-Fock functions

In this paper we consider the distribution of the zeros of a real random Bargmann-Fock function of one or more variables. For these random functions we prove estimates for two types of families of events, both of which are large deviations from the mean. First, we prove that the probability there are no zeros in $[-r,r]^m\subset\R^m$ decays at least exponentially in terms of $r^m$. For this event we also prove a lower bound on the order of decay, which we do not expect to be sharp. Secondly, we compute the order of decay for the probability of families of events where the volume of the complex zero set is either much larger or much smaller then expected.

math.CV↗

arXiv2008

Overcrowding and hole probabilities for random zeros on complex manifolds

We give asymptotic large deviations estimates for the volume inside a domain U of the zero set of a random polynomial of degree N, or more generally, of a holomorphic section of the N-th power of a positive line bundle on a compact Kaehler manifold. In particular, we show that for all $δ>0$, the probability that this volume differs by more than $δN$ from its average value is less than $\exp(-C_{δ,U}N^{m+1})$, for some constant $C_{δ,U}>0$. As a consequence, the "hole probability" that a random section does not vanish in U has an upper bound of the form $\exp(-C_{U}N^{m+1})$.

math.CV↗

arXiv2007

The order of the decay of the hole probability for Gaussian random SU(m+1) polynomials

We show that for Gaussian random SU(m+1) polynomials of a large degree N the probability that there are no zeros in the disk of radius r is less than $e^{-c_{1,r} N^{m+1}}$, and is also greater than $e^{-c_{2,r} N^{m+1}}$. Enroute to this result, we also derive a more general result: probability estimates for the event where the volume of the zero set of a random polynomial of high degree deviates significantly from its mean.

math.CV↗

arXiv2006

The hole probability for Gaussian random SU(2) polynomials

We show that for Gaussian random SU(2)polynomials of a large degree $N$ the probability that there are no zeros in the disk of radius $r$ is less than $e^{-c_{1,r} N^2}$, and is also greater than $e^{-c_{2,r} N^2}$. Enroute to this result, we also derive a more general result: probability estimates for the event that the number of complex zeros of a random polynomial of high degree deviates significantly from its mean.

math.CV↗

arXiv2006

The zeros of Gaussian random holomorphic functions on $\C^n$, and hole probability

We consider a class of Gaussian random holomorphic functions, whose expected zero set is uniformly distributed over $\C^n $. This class is unique (up to multiplication by a non zero holomorphic function), and is closely related to a Gaussian field over a Hilbert space of holomorphic functions on the reduced Heisenberg group. For a fixed random function of this class, we show that the probability that there are no zeros in a ball of large radius, is less than $e^{-c_1 r^{2n+2}}$, and is also greater than $e^{-c_2 r^{2n+2}}$. Enroute to this result we also compute probability estimates for the event that a random function's unintegrated counting function deviates significantly from its mean.

math.CV↗