High–low temperature dualities for the classical β-ensembles

The loop equations for the [Formula: see text]-ensembles are conventionally solved in terms of a [Formula: see text] expansion. We observe that it is also possible to fix [Formula: see text] and expand in inverse powers of [Formula: see text]. At leading order, for the one-point function [Formula: see text] corresponding to the average of the linear statistic [Formula: see text] and after specialising to 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 [Formula: see text] in the case of classical weights — which are particular Riccati equations — are simply related to the differential equations satisfied by [Formula: see text] in the high temperature scaled limit [Formula: see text] ([Formula: see text] fixed, [Formula: see text]), implying a certain high–low temperature duality. A generalisation of this duality, valid without any limiting procedure, is shown to hold for [Formula: see text] and all its higher point analogues in the classical [Formula: see text]-ensembles.