Exact persistence exponent for the $2d$-diffusion equation and related Kac polynomials

TL;DR

This paper calculates the probability decay of sign changes in a 2D diffusion field with random initial conditions, links it to Kac polynomials and random matrices, and explores related zero-crossing properties and spin chain dynamics.

Contribution

It provides the exact persistence exponent for the 2D diffusion equation and establishes new connections with Kac polynomials and random matrix ensembles.

Findings

Persistence exponent for 2D diffusion: 3/16

Decay of Kac polynomial roots probability: n^{-3/4}

Connections with random matrices and spin chain dynamics

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 ${\bf x}$ in the plane, has not changed sign up to time $t$. For large $t$, we show that $p_0(t) \sim t^{-\theta(2)}$ with $\theta(2) = 3/16$. Using the connection between the $2d$-diffusion equation and Kac random polynomials, we show that the probability $q_0(n)$ that Kac polynomials, of (even) degree $n$, have no real root decays, for large $n$, as $q_0(n) \sim n^{-3/4}$. 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 polynomials. Finally, we unveil a precise connection with a fourth model: the semi-infinite Ising spin chain with Glauber dynamics at zero temperature.