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

Abstract

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.

19 pages

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