论文标题
Wigner矩阵产品的特征多项式：有限-N结果和Lyapunov普遍性
Characteristic polynomials of products of Wigner matrices: finite-N results and Lyapunov universality
论文作者
论文摘要
我们计算了$ m $ $ $ $ $ n \ times n $的平均特征多项式，以及$ m $ m $ m $的wigner矩阵的特征多项式的平均值。令人惊讶的是，结果与有限的$ n $的$ m $ $ $ $ $ $ $ $ M $或复杂Ginibre矩阵的产品一致，其中具有I.I.D。高斯条目。对于后者，平均特征多项式产生了乘积矩阵的奇异值的正交多项式，而两个特征多项式的乘积涉及复杂特征值的核。这扩展了一个随机矩阵的一个特征多项式的Forrester和Gamburd的结果，仅取决于前两个矩。在固定$ n $的限制$ m \ to \ infty $中，我们确定了单个特征多项式的零位置，通过取$ m $ th root的对数来重新缩放为lyapunov指数。 $ j $ th零的位置与$ j $ j $渐近同意，与高斯随机矩阵产品的$ j $ th lyapunov exponent的位置暗示了后者的普遍性。
We compute the average characteristic polynomial of the hermitised product of $M$ real or complex Wigner matrices of size $N\times N$ and the average of the characteristic polynomial of a product of $M$ such Wigner matrices times the characteristic polynomial of the conjugate matrix. Surprisingly, the results agree with that of the product of $M$ real or complex Ginibre matrices at finite-$N$, which have i.i.d. Gaussian entries. For the latter the average characteristic polynomial yields the orthogonal polynomial for the singular values of the product matrix, whereas the product of the two characteristic polynomials involves the kernel of complex eigenvalues. This extends the result of Forrester and Gamburd for one characteristic polynomial of a single random matrix and only depends on the first two moments. In the limit $M\to\infty$ at fixed $N$ we determine the locations of the zeros of a single characteristic polynomial, rescaled as Lyapunov exponents by taking the logarithm of the $M$th root. The position of the $j$th zero agrees asymptotically for large-$j$ with the position of the $j$th Lyapunov exponent for products of Gaussian random matrices, hinting at the universality of the latter.
