Blaizot Jean-Paul, Nowak Maciej A
IPTh, CEA-Saclay, 91191 Gif-sur-Yvette, France.
Phys Rev E Stat Nonlin Soft Matter Phys. 2010 Nov;82(5 Pt 1):051115. doi: 10.1103/PhysRevE.82.051115. Epub 2010 Nov 11.
We link the appearance of universal kernels in random matrix ensembles to the phenomenon of shock formation in some fluid dynamical equations. Such equations are derived from Dyson's random walks after a proper rescaling of the time. In the case of the gaussian unitary ensemble, on which we focus in this paper, we show that the characteristics polynomials and their inverse evolve according to a viscid Burgers equation with an effective "spectral viscosity" ν(s)=1/2N, where N is the size of the matrices. We relate the edge of the spectrum of eigenvalues to the shock that naturally appears in the Burgers equation for appropriate initial conditions, thereby suggesting a connection between the well-known microscopic universality of random matrix theory and the universal properties of the solution of the Burgers equation in the vicinity of a shock.
我们将随机矩阵系综中通用核的出现与某些流体动力学方程中的激波形成现象联系起来。此类方程是在对时间进行适当重标度后，从戴森随机游走推导得出的。在本文所关注的高斯酉系综的情形下，我们表明特征多项式及其逆按照一个具有有效“谱粘性”(\nu(s)=\frac{1}{2N})的粘性伯格斯方程演化，其中(N)是矩阵的规模。我们将特征值谱的边缘与在适当初始条件下伯格斯方程中自然出现的激波联系起来，从而暗示了随机矩阵理论中著名的微观普适性与激波附近伯格斯方程解的普适性质之间的联系。
