矩形R-变换作为矩形球面积分的极限
Rectangular R-transform as the limit of rectangular spherical integrals
摘要：
本文将矩形自由概率论与球面积分联系起来。通过这种方式，我们证明了类似的，矩形或正方形的非厄米特矩阵，结果Guionnet和Maida证明厄米特矩阵在2005年。更具体地说，我们研究了当$n,m$趋于无穷大时，$\exp[\sqrt{nm}θX_n]$的期望值的对数（除以$n$）的极限，其中$X_n$是$U_n M_n V_m$的实部，$θ$是实数，$M_n$是确定性矩阵的某个$n\times m$，$U_n, V_m$是具有各自大小的独立Haar分布正交或酉矩阵$n\times n$，$m\times m$ .我们证明了当$M_n$的奇异律收敛到概率测度$μ$时，对于足够小的$θ$，这个极限实际上是存在的，并且可以用$μ$的矩形R变换来表示。这给出了这种变换的解释，它线性化了矩形自由卷积，作为对数拉普拉斯变换序列的极限。
英文摘要：
In this paper, we connect rectangular free probability theory and spherical integrals. In this way, we prove the analogue, for rectangular or square non-Hermitian matrices, of a result that Guionnet and Maida proved for Hermitian matrices in 2005. More specifically, we study the limit, as $n,m$ tend to infinity, of the logarithm (divided by $n$) of the expectation of $\exp[\sqrt{nm}θX_n]$, where $X_n$ is the real part of an entry of $U_n M_n V_m$, $θ$ is a real number, $M_n$ is a certain $n\times m$ deterministic matrix and $U_n, V_m$ are independent Haar-distributed orthogonal or unitary matrices with respective sizes $n\times n$, $m\times m$. We prove that when the singular law of $M_n$ converges to a probability measure $μ$, for $θ$ small enough, this limit actually exists and can be expressed with the rectangular R-transform of $μ$. This gives an interpretation of this transform, which linearizes the rectangular free convolution, as the limit of a sequence of log-Laplace transforms.
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