For a pair of coupled rectangular random matrices we consider the squared singular values of their product, which form a determinantal point process. We show that the limiting mean distribution of these squared singular values is described by the second component of the solution to a vector equilibrium problem. This vector equilibrium problem is defined for three measures with an upper constraint on the first measure and an external field on the second measure. We carry out the steepest descent analysis for a 4 $\times$ 4 matrix-valued Riemann-Hilbert problem, which characterizes the correlation kernel and is related to mixed type multiple orthogonal polynomials associated with the modified Bessel functions. A careful study of the vector equilibrium problem, combined with this asymptotic analysis, ultimately leads to the aforementioned convergence result for the limiting mean distribution, an explicit form of the associated spectral curve, as well as local Sine, Meijer-G and Airy universality results for the squared singular values considered.

中文翻译：

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两个耦合随机矩阵的乘积的大 n 限制

对于一对耦合的矩形随机矩阵，我们考虑它们乘积的平方奇异值，这形成了行列式点过程。我们表明这些平方奇异值的极限平均分布由向量平衡问题的解的第二个分量描述。这个向量平衡问题是为三个度量定义的，第一个度量有上限，第二个度量有一个外场。我们对 4 $\times$ 4 矩阵值 Riemann-Hilbert 问题进行了最速下降分析，该问题表征了相关核，并且与与修正贝塞尔函数相关的混合类型多个正交多项式相关。仔细研究向量平衡问题，结合这种渐近分析，