The classical $\beta$-ensembles with $\beta$ proportional to $1/N$: from loop equations to Dyson's disordered chain

In the classical $\beta$-ensembles of random matrix theory, setting $\beta = 2 \alpha/N$ and taking the $N \to \infty$ limit gives a statistical state depending on $\alpha$. Using the loop equations for the classical $\beta$-ensembles, we study the corresponding eigenvalue density, its moments, covariances of monomial linear statistics, and the moments of the leading $1/N$ correction to the density. From earlier literature the limiting eigenvalue density is known to be related to classical functions. Our study gives a unifying mechanism underlying this fact, identifying in particular the Gauss hypergeometric differential equation determining the Stieltjes transform of the limiting density in the Jacobi case. Our characterisation of the moments and covariances of monomial linear statistics is through recurrence relations. Also, we extend recent work which begins with the $\beta$-ensembles in the high temperature limit and constructs a family of tridiagonal matrices referred to as $\alpha$-ensembles, obtaining a random anti-symmetric tridiagonal matrix with i.i.d.~gamma distributed random variables. From this we are able to supplement analytic results obtained by Dyson in the study of the so-called type I disordered chain.