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 $\mathbf{x}$ 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’s 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’s polynomials. Finally, we unveil a precise connection with a fourth model: the semi-infinite Ising spin chain with Glauber dynamics at zero temperature.

Figure 1

Connections between the four models studied here: they are related to the same Gaussian stationary process (GSP) with correlator $c(T)=\mathrm{sech}(T/2)$. Our main result is the exact value of the persistence exponent for this GSP, $\theta (2)=b=3/16$, together with the full statistics of its zero crossings (5)–(7).