# Two point function for critical points of a random plane wave

## TL;DR

This paper investigates the statistical properties of critical points in random plane waves, providing explicit formulas for their expected counts and revealing a surprising scaling law for their second factorial moment.

## Contribution

It computes the one-point and two-point functions for critical points of random plane waves, highlighting novel asymptotic behavior not previously known.

## Key findings

- Expected number of critical points in a region is derived.
- Short-range asymptotics show the second factorial moment scales as the fourth power of the radius.
- Reveals unexpected scaling law for critical point distribution.

## Abstract

Random plane wave is conjectured to be a universal model for high-energy eigenfunctions of the Laplace operator on generic compact Riemanian manifolds. This is known to be true on average. In the present paper we discuss one of important geometric observable: critical points. We first compute one-point function for the critical point process, in particular we compute the expected number of critical points inside any open set. After that we compute the short-range asymptotic behaviour of the two-point function. This gives an unexpected result that the second factorial moment of the number of critical points in a small disc scales as the fourth power of the radius.