arXiv subjects

Boris Hanin

Publications and source records attributed to Boris Hanin.

59 records · Page 4Linked to original sources

arXiv2014

Mean of the $L^\infty$-norm for $L^2$-normalized random waves on compact aperiodic Riemannian manifolds

This article concerns upper bounds for $L^\infty$-norms of random approximate eigenfunctions of the Laplace operator on a compact aperiodic Riemannian manifold $(M,g).$ We study $f_λ$ chosen uniformly at random from the space of $L^2$-normalized linear combinations of Laplace eigenfunctions with eigenvalues in the interval $(λ^2, \lr{λ+1}^2].$ Our main result is that the expected value of $\norm{f_λ}_\infty$ grows at most like $C \sqrt{\log λ}$ as $λ\to \infty$, where $C$ is an explicit constant depending only on the dimension and volume of $(M,g).$ In addition, we obtain concentration of the $L^\infty$-norm around its mean and median and study the analogous problems for Gaussian random waves on $(M,g).$

math-ph↗

arXiv2014

High Frequency Eigenfunction Immersions and Supremum Norms of Random Waves

A compact Riemannian manifold may be immersed into Euclidean space by using high frequency Laplace eigenfunctions. We study the geometry of the manifold viewed as a metric space endowed with the distance function from the ambient Euclidean space. As an application we give a new proof of a result of Burq-Lebeau and others on upper bounds for the sup-norms of random linear combinations of high frequency eigenfunctions.

math.SP↗

arXiv2013

Nodal Sets of Random Eigenfunctions for the Isotropic Harmonic Oscillator

We consider Gaussian random eigenfunctions (Hermite functions) of fixed energy level of the isotropic semi-classical Harmonic Oscillator on ${\bf R}^n$. We calculate the expected density of zeros of a random eigenfunction in the semi-classical limit $h \to 0.$ In the allowed region the density is of order $h^{-1},$ while in the forbidden region the density is of order $h^{-\frac{1}{2}}$. The computer graphics due to E.J. Heller illustrate this difference in "frequency" between the allowed and forbidden nodal sets.

math.PR↗

arXiv2013

Pairing of Zeros and Critical Points for Random Meromorphic Functions on Riemann Surfaces

We prove that zeros and critical points of a random polynomial $p_N$ of degree $N$ in one complex variable appear in pairs. More precisely, if $p_N$ is conditioned to have $p_N(ξ)=0$ for a fixed $ξ\in \C\backslash\set{0},$ we prove that there is a unique critical point z in the annulus $N^{-1-\ep}<\abs{z-ξ}< N^{-1+\ep}}$ and no critical points closer to $ξ$ with probability at least $1-O(N^{-3/2+3\ep}).$ We also prove an analogous statement in the more general setting of random meromorphic functions on a closed Riemann surface.

math.CV↗

arXiv2012

Correlations and Pairing Between Zeros and Critical Points of Gaussian Random Polynomials

We study the asymptotics of correlations and nearest neighbor spacings between zeros and holomorphic critical points of $p_N$, a degree N Hermitian Gaussian random polynomial in the sense of Shiffman and Zeldtich, as N goes to infinity. By holomorphic critical point we mean a solution to the equation $\frac{d}{dz}p_N(z)=0.$ Our principal result is an explicit asymptotic formula for the local scaling limit of $\E{Z_{p_N}\wedge C_{p_N}},$ the expected joint intensity of zeros and critical points, around any point on the Riemann sphere. Here $Z_{p_N}$ and $C_{p_N}$ are the currents of integration (i.e. counting measures) over the zeros and critical points of $p_N$, respectively. We prove that correlations between zeros and critical points are short range, decaying like $e^{-N\abs{z-w}^2}.$ With $\abs{z-w}$ on the order of $N^{-1/2},$ however, $\E{Z_{p_N}\wedge C_{p_N}}(z,w)$ is sharply peaked near $z=w,$ causing zeros and critical points to appear in rigid pairs. We compute tight bounds on the expected distance and angular dependence between a critical point and its paired zero.

math.PR↗