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首页> 外文期刊>Annales Henri Poincare >Product Matrix Processes for Coupled Multi-Matrix Models and Their Hard Edge Scaling Limits
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Product Matrix Processes for Coupled Multi-Matrix Models and Their Hard Edge Scaling Limits

机译：耦合多矩阵模型的产品矩阵工艺及其硬边缘缩放限制

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Product matrix processes are multi-level point processes formed by the singular values of random matrix products. In this paper, we study such processes where the products of up to m complex random matrices are no longer independent, by introducing a coupling term and potentials for each product. We show that such a process still forms a multi-level determinantal point processes, and give formulae for the relevant correlation functions in terms of the corresponding kernels. For a special choice of potential, leading to a Gaussian coupling between the mth matrix and the product of all previous m-1 matrices, we derive a contour integral representation for the correlation kernels suitable for an asymptotic analysis of large matrix size n. Here, the correlations between the first m-1 levels equal that of the product of m-1 independent matrices, whereas all correlations with the mth level are modified. In the hard edge scaling limit at the origin of the spectra of all products, we find three different asymptotic regimes. The first regime corresponding to weak coupling agrees with the multi-level process for the product of m independent complex Gaussian matrices for all levels, including the m-th. This process was introduced by one of the authors and can be understood as a multi-level extension of the Meijer G-kernel introduced by Kuijlaars and Zhang. In the second asymptotic regime at strong coupling the point process on level m collapses onto level m-1, thus leading to the process of m-1 independent matrices. Finally, in an intermediate regime where the coupling is proportional to n(1/2), we obtain a family of parameter-dependent kernels, interpolating between the limiting processes in the weak and strong coupling regime. These findings generalise previous results of the authors and their coworkers for m=2.
机译：产品矩阵工艺是由随机矩阵产品的奇异值形成的多级点过程。在本文中，我们通过引入每种产品的耦合项和电位，研究多于M个复随机矩阵的产品不再独立的过程。我们表明这种过程仍然形成多级别的确定点过程，并在相应的内核方面给出相关相关函数的公式。对于特殊选择的潜力，导致MTH矩阵与所有先前M-1矩阵的产品之间的高斯耦合，我们推导了适用于大矩阵尺寸N的渐近分析的相关核的轮廓整体表示。这里，第一M-1级之间的相关性等于M-1独立矩阵的乘积，而与MTH级别的所有相关性被修改。在所有产品的Spectra的起源的硬边缩放限制中，我们发现三种不同的渐近制度。对应于弱耦合的第一个制度与M个独立复杂高斯矩阵乘积的多级过程同意，包括所有级别的M-TH。该过程由其中一位作者引入，并且可以理解为Kuijlaars和Zhang引入的Meijer G-Kernel的多级扩展。在强耦合的第二渐近状态下，级别M的点过程折叠到电平M-1上，从而导致M-1独立矩阵的过程。最后，在耦合与N（1/2）成比例的中间区域中，我们获得一系列参数依赖性内核，在弱和强耦合方案中的限制过程之间插值。这些调查结果概括了提交人的先前结果及其同事的M = 2。

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