Related papers: The hole probability for Gaussian random SU(2) pol…
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…
In this paper, we study hole probabilities $P_{0,m}(r,N)$ of SU(m+1) Gaussian random polynomials of degree $N$ over a polydisc $(D(0,r))^m$. When $r\geq1$, we find asymptotic formulas and decay rate of $\log{P_{0,m}(r,N)}$. In dimension…
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…
By a hole we mean a disc that contains no flat chaotic analytic zero points (i.e. zeroes of a random entire function whose Taylor coefficients are independent complex-valued Gaussian variables, and the variance of the k-th coefficient is…
We study the hole probability of Gaussian random entire functions. More specifically, we work with entire functions in Taylor series form with i.i.d complex Gaussian coefficients. A hole is the event where the function has no zeros in a…
We study the hole probability of Gaussian entire functions. More specifically, we work with entire functions in Taylor series form with i.i.d complex Gaussian random variables and arbitrary non-random coefficients. A hole is the event where…
We study the hole probability of Gaussian random entire functions. More specifically, we work with the flat model (the zero set of this function has a distribution which is invariant with respect to the plane isometries). A hole is the…
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…
We study a family of random Taylor series $$F(z) = \sum_{n\ge 0} \zeta_n a_n z^n$$ with radius of convergence almost surely $1$ and independent identically distributed complex Gaussian coefficients $(\zeta_n)$; these Taylor series are…
Consider a polynomial of large degree n whose coefficients are independent, identically distributed, nondegenerate random variables having zero mean and finite moments of all orders. We show that such a polynomial has exactly k real zeros…
Let p_N be a random degree N polynomial in one complex variable whose zeros are chosen independently from a fixed probability measure mu on the Riemann sphere S^2. This article proves that if we condition p_N to have a zero at some fixed…
The "hole probability" that the zero set of the time dependent planar Gaussian analytic function f(z,t) = sum_(n=0)^infty a_n(t) z^n/sqrt(n!), where a_n(t) are i.i.d. complex valued Ornstein-Uhlenbeck processes, does not intersect a disk of…
We investigate the probability that a random polynomial with independent, mean-zero and finite variance coefficients has no real zeros. Specifically, we consider a random polynomial of degree $2n$ with coefficients given by an i.i.d.…
The Kac polynomial $$f_n(x) = \sum_{i=0}^{n} \xi_i x^i$$ with independent coefficients of variance 1 is one of the most studied models of random polynomials. It is well-known that the empirical measure of the roots converges to the uniform…
In this note, we will get the estimate of the expected distribution of critical values of Gaussian SU(2) random polynomials as the degree large enough. The result is a direct application of the Kac-Rice formula. The critical values will…
Let $p(z)=a_0+a_1z+a_2z^2+a_3z^3+\cdots+a_nz^n$ be a polynomial of degree $n,$ where the coefficients $a_j,$ $j \in \{0,1,2,\cdots n\},$ may be complex. We impose some restriction on the coefficients of the real part of the given polynomial…
We give sufficient conditions under which a polyanalytic polynomial of degree $n$ has (i) at least one zero, and (ii) finitely many zeros. In the latter case, we prove that the number of zeros is bounded by $n^2$. We then show that for all…
Given a probability measure $\mu$ with infinite support on the unit circle $\partial\mathbb{D}=\{z:|z|=1\}$, we consider a sequence of paraorthogonal polynomials $\h_n(z,\lambda)$ vanishing at $z=\lambda$ where $\lambda \in \T$ is fixed. We…
Consider a random system $\mathfrak{f}_1(x)=0,\ldots,\mathfrak{f}_n(x)=0$ of $n$ random real polynomials in $n$ variables, where each $\mathfrak{f}_k$ has a prescribed set of exponent vectors in a set $A_k\subseteq \mathbb{Z}^n$ of size…
We consider eigenvalues of a product of n non-Hermitian, independent random matrices. Each matrix in this product is of size N\times N with independent standard complex Gaussian variables. The eigenvalues of such a product form a…