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I'm seeking a probability density function $\Lambda(n,m)$ (where $m$ is the random variable and $n$ is a parameter) describing the probabilities shown in the matrix below, which have been found through simulation: $\Lambda(n,m)$ is the probability that an $n\times n$ real matrix has precisely $m\in\{0,1,\dots,n\}$ eigenvalues with negative real part. I was hoping someone might be able to recognize the pattern:

\begin{pmatrix} 0.5000 & 0.2500 & 0.1045 & 0.0372 & 0.0112 & 0.0029 & 0.0006 \\ 0.5000 & 0.5000 & 0.3955 & 0.2500 & 0.1303 & 0.0568 & 0.0209 \\ & 0.2500 & 0.3955 & 0.4256 & 0.3584 & 0.2471 & 0.1425 \\ & & 0.1045 & 0.2500 & 0.3586 & 0.3865 & 0.3360 \\ & & & 0.0372 & 0.1302 & 0.2471 & 0.3360 \\ & & & & 0.0113 & 0.0568 & 0.1425 \\ & & & & & 0.0029 & 0.0209 \\ & & & & & & 0.0006 \end{pmatrix} where the $(m+1,n)$-th entry is $\Lambda(n,m)$ ($m+1$ because $m$ starts from $0$) and where the blank entries are zeros.

Does anyone recognize this as some density function?


Here is what I've found so far:

  • Since $\Lambda(n,m)$ is a pdf over $m$, $\sum_{m=0}^n \Lambda(n,m)=1.$
  • Symmetry: $\Lambda(n,m)=\Lambda(n,n-m).$
  • I'm fairly certain that $\Lambda(n,0)=\Lambda(n,n)$ is a Gaussian (I can provide details if there is interest).
  • When $m\not=0,n,$ $\Lambda(n,m)$ can be approximated by a Gaussian, where the approximation gets worse the longer away $m$ gets from any of the two endpoints.

Here are some logplots of $\Lambda(n,m)$ as a function of $n$ and then $m$:

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Below is the first plot, but with Gaussians fits shown as well: One can see that the fits gets worse the further $m$ is from either $0$ or $n$:

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