Differential Equations for Dyson Processes
Differential Equations for Dyson Processes
复制标题
戴森过程的微分方程
DOI：
10.1007/s00220-004-1182-8
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发表时间：
2003
影响因子：
2.4
通讯作者：
H. Widom
中科院分区：
文献类型：
--
作者：
C. Tracy;H. Widom
Dyson过程是矩阵系综上的一个过程，其中的元素是扩散的。我们感兴趣的是这种矩阵的特征值（或奇异值）的分布。在最初的Dyson过程中，它是n×n个Hermitian矩阵的集合，特征值描述n条曲线。给定集合X1，...，Xm对于每个k，在时间τk没有曲线通过Xk的概率由某个矩阵核（扩展的Hermite核）的Fredholm行列式给出。因此，我们称这种戴森过程为厄米过程。类似地，当一个复矩阵的元素经历扩散时，我们称其奇异值的演化为拉盖尔过程，对于它有一个相应的扩展拉盖尔核。在边缘处按比例缩放Hermite过程导致Airy过程（这是由Prähofer和Spohn作为多核生长模型的极限平稳过程引入的），在整体上导致sine过程;在边缘处按比例缩放Laguerre过程导致Bessel过程。在早期的工作中，作者发现了一个具有独立变量的常微分方程组，其解决定了概率
其中τ→A（τ）表示艾里过程的顶曲线。我们的第一个结果是对这一点的概括和加强。我们假设每个Xk是一个有限的区间并，并找到一个偏微分方程组，以Xk的区间端点为自变量，其解决定了对于每个k，在时间τk没有曲线通过Xk的概率。然后，我们发现类似的系统的厄米过程（这是更复杂的），也为正弦过程。最后，我们找到一个类似的系统的偏微分方程的贝塞尔过程，这是最困难的。
AbstractWe call a Dyson process any process on ensembles of matrices in which the entries undergo diffusion. We are interested in the distribution of the eigenvalues (or singular values) of such matrices. In the original Dyson process it was the ensemble of n×n Hermitian matrices, and the eigenvalues describe n curves. Given sets X1,...,Xm the probability that for each k no curve passes through Xk at time τk is given by the Fredholm determinant of a certain matrix kernel, the extended Hermite kernel. For this reason we call this Dyson process the Hermite process. Similarly, when the entries of a complex matrix undergo diffusion we call the evolution of its singular values the Laguerre process, for which there is a corresponding extended Laguerre kernel. Scaling the Hermite process at the edge leads to the Airy process (which was introduced by Prähofer and Spohn as the limiting stationary process for a polynuclear growth model) and in the bulk to the sine process; scaling the Laguerre process at the edge leads to the Bessel process.In earlier work the authors found a system of ordinary differential equations with independent variable ξ whose solution determined the probabilities
where τ→A(τ) denotes the top curve of the Airy process. Our first result is a generalization and strengthening of this. We assume that each Xk is a finite union of intervals and find a system of partial differential equations, with the end-points of the intervals of the Xk as independent variables, whose solution determines the probability that for each k no curve passes through Xk at time τk. Then we find the analogous systems for the Hermite process (which is more complicated) and also for the sine process. Finally we find an analogous system of PDEs for the Bessel process, which is the most difficult.