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I have posted this to MSE, but it got no attention (https://math.stackexchange.com/questions/2619324/average-of-tracy-widom-distributions)

The Tracy-Widom distributions famously describe the statistics of the largest eigenvalue of gaussian random matrices. There are 3 of them, labeled by $\beta$ which is 1, 2 or 4 for real, complex or quaternion matrices, respectively.

If the matrices are distributed with density $$e^{-\frac{\beta}{2}{\rm Tr}H^2},$$ the average of the largest eigenvalue $\lambda$ is $\sqrt{2N}$ for all $\beta$ and the distribution around the average is Tracy-Widom $F_\beta$. That is, defining $$\lambda=\sqrt{2N}+\frac{x}{\sqrt{2}N^{1/6}}$$ then $x$ has dentsity $F_\beta(x)$.

What I don't understand is why the distribution $F_\beta(x)$ does not have zero average. As can be seen in page 8 of this nice review by Tracy and Widom themselves, https://arxiv.org/pdf/solv-int/9707001.pdf, the averages are approximately -1.20, -1.77 and -2.30 for $\beta=1,2,4$.

Shoudn't the variable $x$ have zero average by definition?

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    $\begingroup$ no, the average of $\lambda$ is only $\sqrt{2N}$ in the large-$N$ limit, there are order $N^{-1/6}$ corrections to $E(\lambda)$ which are described by the nonzero average of $x$. $\endgroup$
    – Carlo Beenakker
    Commented Jan 26, 2018 at 12:41
  • $\begingroup$ oh, of course. that was silly... $\endgroup$
    – thedude
    Commented Jan 26, 2018 at 14:19

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