Configurational Temperature in Matrix Models and Random Matrix Ensembles

We investigate the configurational temperature estimator in interacting matrix models and Gaussian random-matrix ensembles. The estimator follows from an exact Schwinger--Dyson identity and may be expressed in terms of the gradient and Hessian of the action. We study the Gross--Witten--Wadia model, a quartic double-well matrix model, and the Gaussian Orthogonal, Unitary, and Symplectic Ensembles. In all cases, the estimator satisfies the exact Schwinger--Dyson identity, $\beta_{\rm config} = 1$, within statistical uncertainties. Separating the estimator into isotropic and anisotropic parts, we find that the leading finite-$N$ corrections satisfy the approximate relation $\beta_{\rm iso} - 1 \simeq - \beta_{\rm aniso}$. We also show that the configurational temperature estimator provides a sensitive diagnostic of Monte Carlo simulations.