title: Statistics of the number of zero crossings: from random polynomials to the 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(-theta(d)) where theta(d)>0 is the exponent associated with 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(-phi(k/logn)) where phi(x) is a universal large deviation function.
authors.affiliation: Laboratoire de Physique Théorique (UMR du CNRS 8627), Université de Paris-Sud, 91405 Orsay Cedex, France.
authors.forename: Grégory
authors.initials: G
authors.lastname: Schehr
authors.forename: Satya N
authors.initials: SN
authors.lastname: Majumdar