Refined evaluation of the level-spacing distribution of symplectic ensembles: moments and implications

To obtain a more precise value for the variance sigma2 of the joint probability distribution of a symplectic ensemble, we extend previous numerical evaluations of a power series. Our result sigma2 approximately 0.1041 shows that the excellent approximation using the analytically simple Wigner surmise fractionally overestimates this value. This behavior is important in establishing the trend of a generalization of the surmise to describe the terrace-width distribution on vicinal surfaces. We also obtain precise estimates of the skewness and the kurtosis of the exact distribution, as well as the related moments.