High-low temperature dualities for the classical β-ensembles

Abstract

The loop equations for the β-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 β. At leading order, for the one-point function W1(x) corresponding to the average of the linear statistic A =aj=1N1/(x-λ j) 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 W1(x) in the case of classical weights-which are particular Riccati equations-are simply related to the differential..