  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.
