Recently I got interested in the world of Random Matrix Models and I bumped into some generalizations of the usual random matrix theories classified by Dyson whose probability density functions are:
$P(\lambda_1, \dots, \lambda_n) = \frac{1}{Z_{N, \beta}} \prod_{i<j}|\lambda_i-\lambda_j|^{\beta} e^{-N \sum_{k=1}^{N} V(\lambda_k)} $
where $Z_{N, \beta}$ is a normalization constant and $\beta =1, 2, 4$ in the Dyson classification. In the generalizations I'm interested into $\beta$ is allowed to assume every real value $\beta>0$ and in particular if $V(x)=\frac{1}{2} x^2$ these ensembles are called "$\beta$-Hermite ensembles"
My question is about how and if some universal features of the classical Dyson ensembles persist after very little perturbations of $\beta$ away from its more common values. In particular it is well know that in the Dyson $\beta=2$ ensemble we are expected to find a universal "sine Kernel" behaviour when considering the density-density correlator $\langle \rho(\lambda_1) \rho(\lambda_2) \rangle$ with $\lambda_1$ and $\lambda_2$ sufficiently "close" to one another. I'm wondering if it is true that by considering $\beta= 2 + \epsilon$ with $\epsilon$ an "infinitesimal parameter" I can expect to find a similar behaviour for the correlator with only a small deviation of order $\epsilon$ away from the "Sine Kernel" behaviour or if even the slightest deformation can determine a completely different behaviour of the correlator above.
I had a look into the literature but I failed to find a paper addressing this topic. Thank you for your help!
To clarify the notation that I used: $\rho(\lambda)= \sum_{k=1}^{N} \delta(\lambda- \lambda_k)$.