Mathematical Sciences: Research in Random Matrices and Spectral Asymptotics

Widom DMS-9424292 Widom已经证明，存在与重要的随机矩阵模型相关的微分方程，并利用这一事实获得了理论中出现的各种量的显式表达式。 他将继续这项调查，以确定在何种程度上适用于满足一些共同条件的模型类的结果。 一类确定性问题涉及有限Toeplitz矩阵及其类似矩阵特征值的渐近性。 每一个都有一个相关的“符号”，人们期望将特征值的分布与符号的值联系起来。 这个建议涉及的应用算子理论的随机矩阵模型。 在随机矩阵理论中，自然产生的基本量通常与某些算子的行列式有关。 这些行列式的渐近行为在算子理论中具有独立的意义。

Widom DMS-9424292 Widom has shown that there are differential equations associated with important random matrix models, and has used this fact to obtain explicit expressions for various quantities arising in the theory. He will continue this investigation to determine the extent to which the results apply to classes of models satisfying some common conditions. One class of deterministic problems deals with the asymptotics of the eigenvalues of finite Toeplitz matrices and their analogues. Each of these has an associated "symbol," and one expects to relate the distribution of eigenvalues to the values of the symbol. This proposal concerns the applications of operator theory to random matrix models. In random matrix theory fundamental quantities that arise naturally are often related to determinants of certain operators. The asymptotic behavior of these determinants is of independent interest in operator theory.