论文标题
$ m $ $ $ $ n \ times n $ ginibre矩阵的产品：关键制度中的真实特征值$ m = o（n）$
The Product of $m$ real $N\times N$ Ginibre matrices: Real eigenvalues in the critical regime $m=O(N)$
论文作者
论文摘要
我们研究产品$ p_m $ $ m $ $ $ $ $ $ n $的高斯元素，最近收到了新的利息。它的特征值是真实的，要么是复杂的共轭对，在$ m \ to \ ind $ n $时，概率是真实的。在这种制度中，统计数据成为确定性，而Lyapunov频谱已很久以前得出。另一方面，当固定$ n \ to \ infty $和$ m $固定时，可以预期，与原始统计数据相同的本地统计数据与单个真实的ginibre合奏相同，$ m = 1 $。受复杂吉尼伯矩阵产物的类似发现的启发，当两个参数成比例时，我们引入了临界缩放制度，$ m =αn$。在这个关键方案中，我们得出了实际特征值的预期数量，方差和重新确定的密度。这使我们能够分别分别在$α\ to \ infty $和$α\ to0 $的情况下，在上述限制中的最新结果之间进行插值。
We study the product $P_m$ of $m$ real Ginibre matrices with Gaussian elements of size $N$, which has received renewed interest recently. Its eigenvalues, which are either real or come in complex conjugate pairs, become all real with probability one when $m\to\infty$ at fixed $N$. In this regime the statistics becomes deterministic and the Lyapunov spectrum has been derived long ago. On the other hand, when $N\to\infty$ and $m$ is fixed, it can be expected that away from the origin the same local statistics as for a single real Ginibre ensemble at $m=1$ prevails. Inspired by analogous findings for products of complex Ginibre matrices, we introduce a critical scaling regime when the two parameters are proportional, $m=αN$. We derive the expected number, variance and rescaled density of real eigenvalues in this critical regime. This allows us to interpolate between previous recent results in the above mentioned limits when $α\to\infty$ and $α\to0$, respectively.
