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 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 $W_1(x)$ 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.

的循环方程$\beta$- 集成通常根据$1/ñ$扩张。我们观察到也可以修复$ñ$并展开$\beta$. 以领先顺序，对于单点函数$W_1(X)$对应于线性统计量的平均值$\mathrm{一种}={\sum }_{j=1}^{ñ}1/(X-{\lambda }_j)$在专门研究经典权重之后，这恢复了 Stieltjes 将经典多项式的零点与某些对数气体势能的最小能量配置相关联的众所周知的结果。此外，观察到微分方程满足$W_1(X)$在经典权重的情况下——它们是特定的 Riccati 方程——简单地与满足的微分方程有关$W_1(X)$在高温标度极限$\beta =2\alpha /ñ$($\alpha$固定的，$ñ\to \infty$)，暗示了一定的高低温二元性。这种对偶性的概括，没有任何限制程序有效，被证明适用于$W_1(X)$以及经典中所有更高点的类似物$\beta$- 合奏。