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For a physics application (quantum delay times of a chaotic scatterer) I need to generate $m$ positive random variables $\lambda_1,\lambda_2,\ldots\lambda_m$ with probability distribution $$P_\beta(\lambda_1,\lambda_2,\ldots\lambda_m)\propto \prod_{k=1}^{m}\lambda_k^{\beta m/2}e^{-\beta\lambda_k/2}\prod_{1\leq i<j\leq m}|\lambda_i-\lambda_j|^\beta,\;\;\lambda_k\geq 0.\qquad [1]$$ The index $\beta\in\{1,2,4\}$.

An efficient way to generate these random variables, discussed for example in Forrester's book, is to use the eigenvalues $\mu_1,\mu_2,\ldots\mu_m$ of Wishart matrices $W^\dagger W$, constructed from a real ($\beta=1$), complex ($\beta=2$) or quaternion ($\beta=4$) matrix $W$ of dimension $n\times m$ (with $n\geq m$). If the matrix elements are i.i.d. from a normal distribution, the eigenvalue distribution is

$$P_\beta(\mu_1,\mu_2,\ldots\mu_m)\propto \prod_{k=1}^{m}\mu_k^{\beta a/2}e^{-\beta\mu_k/2}\prod_{1\leq i<j\leq m}|\mu_i-\mu_j|^\beta,\;\;\mu_k\geq 0,\qquad [2]$$

with $a=n-m+1-2/\beta$.

I can identify the distributions [1] and [2] by equating $a=m\Rightarrow n=2m-1+2/\beta$. This works for $\beta=1$ and for $\beta=2$, but fails for $\beta=4$, because it would imply a half-integer dimension of $W$.

Question: Is there a work-around which would allow me to generate the distribution [1] for $\beta=4$ using Wishart matrices? I'm basically asking for a way to make sense of "half a quaternion" --- perhaps I'm asking too much and the answer is just "no way".

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This is not an answer to trying to define a half quaternion; instead,the construction of a random matrix model for arbitrary $a> \beta(m-1)/2$ and arbitrary $\beta> 0$ by using tri-diagonal matrices (generalizing Wishart distributions) was done by Dumitiru and Edelman: http://www.math.washington.edu/~dumitriu/JMathPhys_43_5830.pdf

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  • $\begingroup$ yes, that does the trick, thank you very much! $\endgroup$ – Carlo Beenakker Dec 2 '14 at 20:35

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