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 the Gaudin-Mehta 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?

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?