Boris Hanin

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

Scaling limit for the kernel of the spectral projector and remainder estimates in the pointwise Weyl law

p_N

Pairing of zeros and critical points for random polynomials

, 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

Fixed frequency eigenfunction immersions and supremum norms of random waves

\frac{d}{dz}p_N(z)=0.

Pairing of zeros and critical points for random meromorphic functions on Riemann surfaces

Our principal result is an explicit asymptotic formula for the local scaling limit of

Scaling of Harmonic Oscillator Eigenfunctions and Their Nodal Sets Around the Caustic

\E{Z_{p_N}\wedge C_{p_N}},

Nodal Sets of Random Eigenfunctions for the Isotropic Harmonic Oscillator

the expected joint intensity of zeros and critical points, around any point on the Riemann sphere. Here

Universal Function Approximation by Deep Neural Nets with Bounded Width and ReLU Activations.

Z_{p_N}

Approximating Continuous Functions by ReLU Nets of Minimal Width.

and

Local Integral Statistics for Monochromatic Random Waves

C_{p_N}