大型随机矩阵的无穷乘积与矩阵值扩散
Infinite Products of Large Random Matrices and Matrix-valued Diffusion
摘要：
本文利用随机矩阵理论中的一个推广的图解规则来计算大型复矩阵和大型厄米矩阵的有限乘积和无限乘积的谱性质。无限乘积的情况下，我们可以定义一个自然的矩阵值乘法扩散过程。在这两种情况下的厄米和复杂的矩阵，我们观察到出现的“拓扑相变“的频谱，经过一定的临界扩散时间$τ_{\rm crit}$达到。在两个厄米系综的特殊产品的情况下，我们也观察到一个不寻常的局域化-离域相变在所考虑的系综的频谱。我们通过数值模拟验证了在这项工作中得到的解析公式。
英文摘要：
We use an extension of the diagrammatic rules in random matrix theory to evaluate spectral properties of finite and infinite products of large complex matrices and large hermitian matrices. The infinite product case allows us to define a natural matrix-valued multiplicative diffusion process. In both cases of hermitian and complex matrices, we observe an emergence of "topological phase transition" in the spectrum, after some critical diffusion time $τ_{\rm crit}$ is reached. In the case of the particular product of two hermitian ensembles, we observe also an unusual localization-delocalization phase transition in the spectrum of the considered ensemble. We verify the analytical formulae obtained in this work by numerical simulation.
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