Let $M$ come from an ensemble of $N\times N$ matrices. The Wigner surmise is density function $p^W_0(s)=\frac{\pi}{2}se^{-\pi s^2/4}$. From a random matrix point of view, we can write $\rho^W_0(s)=\frac{d^2}{ds^2}E((0,s))$, where $E(I)$ is the eigenvalue gap probability: $M$ has no eigenvalues in interval $I$. $E(I)$ and it's derivatives are intimately related to the correlations between nearest neighbors.

> Question 1: What are known random matrix ensembles which have
> their eigenvalue gap probability *exactly* equal to the Wigner
> surmise? 
> 
> 
> Question 2: What are known interacting-particle systems which have
> their gap probability *exactly* equal to the Wigner surmise?

One [example that I've seen][1] is the real Ginibre ensemble which takes a random Gaussian matrix and focuses only on *real* eigenvalues. Then the probability of there being an *even* number of eigenvalues in $[0,s]$ matches the Wigner surmise. This is equivalent to certain statistics of creation/annihilation processes on the line. In addition to this, there are [some statistical physics spin systems][2] which seem to give an exact surmise as well. Unfortunately I'm not an expert in the latter area.



Some more background:


It's a well known fact that many random matrix ensembles exhibit a density function of the form:

$$p_0(s)=\frac{2u(\pi^2 s^2/4)}{s}\exp\left(-\int_0^{\pi^2 s^2/4}\frac{u(t)}{t}dt\right),$$

where $u$ satisfies a Painleve equation and of which $\rho_0^W(s)$ is a special case. So in short, $\rho^W_0(s)$ is usually an *approximation*, not an exact answer. One can certainly derive some conditions on $u$ and the resulting Painleve equation to get an exact Wigner surmise but this doesn't answer from which random matrix ensembles it comes from.


  [1]: http://arxiv.org/pdf/1306.4106v2.pdf
  [2]: http://arxiv.org/pdf/0712.2011v1.pdf