High-low temperature dualities for the classical $\beta$-ensembles

The loop equations for the $\beta$-ensembles are conventionally solved in terms of a $1/N$ expansion. We observe that it is also possible to fix $N$ and expand in inverse powers of $\beta$. At leading order, for the one-point function $W_1(x)$ corresponding to the average of the linear statistic $A = \sum_{j=1}^N 1/(x - \lambda_j)$, and specialising the classical weights, this reclaims well known results of Stieltjes relating the zeros of the classical polynomials to the minimum energy configuration of certain log-gas potential energies. Moreover, it is observed that the differential equations satisfied by $W_1(x)$ in the case of classical weights -- which are particular Riccati equations -- are simply related to the differential equations satisfied by $W_1(x)$ in the high temperature scaled limit $\beta = 2\alpha/N$ ($\alpha$ fixed, $N \to \infty$), implying a certain high-low temperature duality. A generalisation of this duality, valid without any limiting procedure, is shown to hold for $W_1(x)$ and all its higher point analogues in the classical $\beta$-ensembles.