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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过程及其难点   边缘缩放限制

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Product matrix processes are multi-level point processes formed by thesingular values of random matrix products. In this paper we study suchprocesses where the products of up to $m$ complex random matrices are no longerindependent, by introducing a coupling term and potentials for each product. Weshow that such a process still forms a multi-level determinantal pointprocesses, and give formulae for the relevant correlation functions in terms ofthe corresponding kernels. For a special choice of potential, leading to a Gaussian coupling between the$m$th matrix and the product of all previous $m-1$ matrices, we derive acontour integral representation for the correlation kernels suitable for anasymptotic analysis of large matrix size $n$. Here, the correlations betweenthe first $m-1$ levels equal that of the product of $m-1$ independent matrices,whereas all correlations with the $m$th level are modified. In the hard edgescaling limit at the origin of the spectra of all products we find threedifferent asymptotic regimes. The first regime corresponding to weak couplingagrees with the multi-level process for the product of $m$ independent complexGaussian matrices for all levels, including the $m$-th. This process wasintroduced by one of the authors and can be understood as a multi-levelextension of the Meijer $G$-kernel introduced by Kuijlaars and Zhang. In thesecond asymptotic regime at strong coupling the point process on level $m$collapses onto level $m-1$, thus leading to the process of $m-1$ independentmatrices. Finally, in an intermediate regime where the coupling is proportionalto $n^{\frac12}$, we obtain a family of parameter dependent kernels,interpolating between the limiting processes in the weak and strong couplingregime. These findings generalise previous results of the authors and theircoworkers for $m=2$.
机译：乘积矩阵过程是由随机矩阵乘积的奇异值形成的多级点过程。在本文中，我们通过引入每个产品的耦合项和势，研究了这样的过程，其中高达$ m $的复杂随机矩阵的乘积不再独立。我们展示了这样一个过程仍然形成了一个多层次的行列式点过程，并根据相应的内核给出了相关函数的公式。对于可能的特殊选择，导致第m个矩阵与所有先前$ m-1 $矩阵的乘积之间的高斯耦合，我们导出了适合于大矩阵规模的渐近分析的相关核的等高线积分表示。 n $。在这里，前$ m-1 $个水平之间的相关性等于$ m-1 $个独立矩阵乘积的相关性，而所有与第mm个水平相关的相关性均被修改。在所有产品的光谱起源处的硬边缩放极限中，我们发现了三种不同的渐近状态。对应于弱耦合的第一个机制与$ m $独立复数乘积高斯矩阵的乘积的所有层次（包括第m $ m $）的多级过程相一致。此过程由一位作者介绍，可以理解为Kuijlaars和Zhang引入的Meijer $ G $内核的多级扩展。在第二种渐近状态下，强耦合的点过程在$ m $级上崩溃到$ m-1 $级，从而导致$ m-1 $个独立矩阵的过程。最后，在耦合与$ n ^ {\ frac12} $成比例的中间状态下，我们获得了一组依赖参数的内核，它们在弱耦合和强耦合区域的极限过程之间进行插值。这些发现概括了作者及其同事的先前结果，其中$ m = 2 $。

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