Exact persistence exponent for the \(2d\)-diffusion equation and related Kac polynomials

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.