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

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^{-θ(d)} where θ(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(θ(d) + θ(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 ϕ(k/\log n)} where \tilde ϕ(x) is a universal large deviation function.

4 pages, 3 figures. Minor changes. Accepted version in Phys. Rev. Lett