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Let $M_n$ be drawn from $n\times n$ matrices under the Circular Orthogonal Ensemble (COE) distribution. Then the eigenvalues of $M_n$ all lie on the unit circle. Starting on the real line and going counterclockwise, order the eigenvalues $\lambda_1,\lambda_2,\cdots,\lambda_n$ by their argument, so that moving counterclockwise off the real line, $\lambda_1$ is the first eigenvalue that one stumbles on. The gap probability $p_n(x)$ will be the probability that there are no eigenvalues with argument in the range $[0,x]$, where $x\in[0,2\pi)$. This is equivalent to saying $\arg(\lambda_1)\geq x$. It is well known that $p_n(x)$ converges to a Gaudin-Mehta-like distribution $p(x)$ which looks something like $e^{-cx^2}I(x)$ where $I(x)$ is a special function.

I have a sequence of random variables $X_n$ which I believe converge weakly to $p(x)$. Loosely speaking, $X_n$ comes from a point process of $n$ particles ("eigenvalues"). I'm wondering if there is a method-of-moments proof approach. I haven't seen anything similar done in the literature. I'm not even sure the moments of $p(x)$ satisfy a Carelman condition. To be more specific, usually when one proves results concerning $p(x)$ one has access to the full correlation function of eigenvalues, so that $p(x)$ has a nice expression through Fredholm determinents. I don't have a "full" correlation function, just one that connects a single pair of consecutive (in argument) eigenvalues, say $\lambda_1$ and $\lambda_2$, or just the distribution of $\lambda_1$ (remember, I've ordered them by argument). So, I'm not seeing any nice determinental expressions in my work making the proof of convergence rather unobvious. Has anyone seen similar work done for such convergence?

Usually in work involving say, TASEP processes, one has nice determinental expressions for correlations. So, I would be very interested in seeing convergence to gap probability results that do not have an obvious determinental structure. For example, are there results showing convergence to the correct Painleve equation which $p(x)$ satisfies?

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  • $\begingroup$ it seems to me that $1-p_n(x)$ will vanish linearly with $x$, while the Gaudin-Mehta result vanishes quadratically; the difference arises because $\lambda_1$ is not repelled from the real line; am I mistaken? (note that $M_n$ is not an orthogonal matrix, unlike the name COE suggests, it's a symmetric unitary matrix, so the real axis is not a line of symmetry) $\endgroup$
    – Carlo Beenakker
    Commented Jul 1, 2014 at 10:37
  • $\begingroup$ @CarloBeenakker: I might be mixing up terminology but, for the GUE (and therefore for the CUE), one has the Gaudin Mehta result which looks like $xe^{-cx}I(x)$ whereas for the COE there is no $x$ term out front. I've changed my wording to Gaudin-Mehta-like result just in case. $\endgroup$
    – Alex R.
    Commented Jul 1, 2014 at 16:27
  • $\begingroup$ just to help me, perhaps you can be more specific what you call the "Gaudin-Metha-like" result $p(x)$? Is it the integrated spacing distribution? $\endgroup$
    – Carlo Beenakker
    Commented Jul 1, 2014 at 17:37
  • $\begingroup$ I am not sure that I understand the question, so let me answer a more general one: you can do (some) gap distributions by the method of moments if you work with non-backtracking paths, but this is not sharp in the interior of the support. See this paper of Sodin for a review: arxiv.org/abs/1406.3410 $\endgroup$
    – ofer zeitouni
    Commented Jul 1, 2014 at 19:26

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