Exact Persistence Exponent for the2D-Diffusion Equation and Related Kac Polynomials

Abstract

We compute the persistence for the 2D-diffusion equation with random initial condition, i.e., the probability p_{0}(t) that the diffusion field, at a given point x in the plane, has not changed sign up to time t. For large t, we show that p_{0}(t)∼t^{-θ(2)} with θ(2)=3/16. Using the connection between the 2D-diffusion equation and Kac random polynomials, we show that the probability q_{0}(n) that Kac's polynomials, of (even) degree n, have no real root decays, for large n, as q_{0}(n)∼n^{-3/4}. We obtain this result by using yet another connection with the truncated orthogonal ensemble of random matrices. This allows us to compute various properties of the zero crossings of the diffusing field, equivalently of the real roots of Kac's polynomials. Finally, we unveil a precise connection with a fourth model: the semi-infinite Ising spin chain with Glauber dynamics at zero temperature.