Following the works by Wiegmann-Zabrodin, Elbau-Felder, Hedenmalm-Makarov,and others, we consider the normal matrix model with an arbitrary potentialfunction, and explain how the problem of finding the support domain for theasymptotic eigenvalue density of such matrices (when the size of the matricesgoes to infinity) is related to the problem of Hele-Shaw flows on curvedsurfaces, considered by Entov and the first author in 1990-s. In the case whenthe potential function is the sum of a rotationally invariant function and thereal part of a polynomial of the complex coordinate, we use this relation andthe conformal mapping method developed by Entov and the first author to findthe shape of the support domain explicitly (up to finitely many undeterminedparameters, which are to be found from a finite system of equations). In thecase when the rotationally invariant function is $\beta |z|^2$, this is done byWiegmann-Zabrodin and Elbau-Felder. We apply our results to the generalizednormal matrix model, which deals with random block matrices that give rise to*-representations of the deformed preprojective algebra of the affine quiver oftype $\hat A_{m-1}$. We show that this model is equivalent to the usual normalmatrix model in the large $N$ limit. Thus the conformal mapping method can beapplied to find explicitly the support domain for the generalized normal matrixmodel.
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机译：遵循Wiegmann-Zabrodin，Elbau-Felder，Hedenmalm-Makarov等人的著作，我们考虑具有任意势函数的正态矩阵模型，并解释了如何找到此类矩阵的渐近特征值密度的支持域的问题（当矩阵的大小到无穷大）与Helve-Shaw在曲面上的流动问题有关，由Entov和1990年代的第一作者考虑。在势函数是旋转不变函数与复数坐标多项式的实部之和的情况下，我们使用此关系以及Entov和第一作者开发的共形映射方法来明确地找到支撑域的形状（向上到有限的许多不确定的参数，这些参数可以从有限的方程组中找到。在旋转不变函数为$ \ beta | z | ^ 2 $的情况下，这是由Wiegmann-Zabrodin和Elbau-Felder完成的。我们将我们的结果应用于广义法线矩阵模型，该模型处理随机块矩阵，从而产生类型为\\ hat A_ {m-1} $的仿射箭的变形预投影代数的*-表示。我们表明，此模型在$ N $大限制下等效于通常的normalmatrix模型。因此，可以采用保形映射方法来为广义法线矩阵模型明确找到支持域。
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