Let $\chi^{\mathrm{Ai}}(I)$ denote the GUE (Airy-2) point process on the interval $I \subset \mathbb{R}$. Soshnikov proved \begin{align} \mathbb{E}(\chi^{\mathrm{Ai}}(-T, +\infty)) &\sim \frac{2}{3\pi}T^{3/2} + O(1) \\ \mathrm{Var} \left (\chi^{\mathrm{Ai}}(I_k(T)) \right )&\sim \frac{11}{12\pi^2} \log T+ O(1), \end{align} where \begin{align} I_1(T) &:= (-T, +\infty) \\ I_k(T) &:= (-kT, -(k-1)T). \end{align} Does anyone know if analogous results have been proved for the GOE (Airy-1) point process? I would be quite surprised if not, however I have not been able to find any results like this in the literature.


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According to arXiv:0909.2677 (page 9 and 14), the expectation values for GOE and GUE are the same, while the variance in the GOE is twice that in the GUE.

I also note that, according to J. Hägg's Ph.D thesis (page 5), there is an error in Soshnikov's paper as a result of which the coefficient $11/12\pi^2$ in the variance should be $3/4\pi^2$.

  • $\begingroup$ I believe the results on page 9 and 14 hold for the bulk eigenvalue point process. The GOE and GUE point process refers to the edge-scaled eigenvalue point process. $\endgroup$ – nootnoot Dec 18 '18 at 6:36
  • $\begingroup$ if I look at, for example, lemmas 26 and 27, I see that the limit $t\rightarrow 1$ is taken, which is the limit of the spectral edge; these are edge-scaled expectations and variances. $\endgroup$ – Carlo Beenakker Dec 18 '18 at 9:33
  • $\begingroup$ I see that, but the results on page 9 and 14 are given for the finite counting process, though on arbitrary intervals. Are you saying that a suitable choice of the interval plus proper scaling can recover the GOE/GUE point process? I know that $\{ \frac{\lambda_{N-k} - 2\sqrt{N}}{N^{1/6}} \}_{k=0}^{k_0} \xrightarrow{d} \{a_k \}_{k=0}^{k_0} $, but this convergence is finite dimensional. $\endgroup$ – nootnoot Dec 18 '18 at 19:19

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