King's College London, Department of Mathematics, London WC2R 2LS, United Kingdom.
LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France.
Phys Rev Lett. 2018 Oct 12;121(15):150601. doi: 10.1103/PhysRevLett.121.150601.
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.
我们计算了具有随机初始条件的 2D-扩散方程的持久性，即在平面上给定点 x 的扩散场在时间 t 之前没有改变符号的概率 p_{0}(t)。对于大的 t，我们表明 p_{0}(t)∼t^{-θ(2)}，其中 θ(2)=3/16。利用 2D-扩散方程与 Kac 随机多项式之间的联系，我们表明 Kac 多项式（偶数）度 n 的无实根概率 q_{0}(n)，对于大的 n，q_{0}(n)∼n^{-3/4}。我们通过与随机矩阵截断正交系的另一个联系得到了这个结果。这使我们能够计算扩散场的零点交叉（即 Kac 多项式的实根）的各种性质。最后，我们揭示了与第四个模型的精确联系：零温度下的 Glauber 动力学的半无限伊辛自旋链。
