Statistics of the Number of Zero Crossings : from Random Polynomials to Diffusion Equation

TL;DR

This paper explores the statistical properties of real roots in large-degree random polynomials and links their behavior to diffusion equations with random initial conditions, revealing universal scaling laws.

Contribution

It establishes a connection between the number of real roots of random polynomials and the persistence probabilities of diffusion equations, introducing universal large deviation functions.

Findings

Probability of no roots in [0,1] decays as a power law in degree n

Exact number of roots follows a universal large deviation scaling

Connection to diffusion equations provides physical insights

Abstract

We consider a class of real random polynomials, indexed by an integer d, of large degree n and focus on the number of real roots of such random polynomials. The probability that such polynomials have no real root in the interval [0,1] decays as a power law n^{-\theta(d)} where \theta(d)>0 is the exponent associated to the decay of the persistence probability for the diffusion equation with random initial conditions in space dimension d. For n even, the probability that such polynomials have no root on the full real axis decays as n^{-2(\theta(d) + \theta(2))}. For d=1, this connection allows for a physical realization of real random polynomials. We further show that the probability that such polynomials have exactly k real roots in [0,1] has an unusual scaling form given by n^{-\tilde \phi(k/\log n)} where \tilde \phi(x) is a universal large deviation function.