Accuracy and range of validity of the Wigner surmise for mixed symmetry classes in random matrix theory

TL;DR

This paper evaluates the accuracy of the Wigner surmise in approximating level spacing distributions during the transition between orthogonal and unitary random matrix ensembles, confirming its validity for small to moderate spacings.

Contribution

It provides a detailed comparison between the Wigner surmise and large-N results for the crossover between Gaussian orthogonal and unitary ensembles, establishing the surmise's range of validity.

Findings

Wigner surmise accurately approximates level spacing distributions for 0  s  2

The surmise's validity is confirmed for small to moderate spacings

Comparison with large-N results shows good agreement within the validity range

Abstract

Schierenberg et al. [Phys. Rev. E 85, 061130 (2012)] recently applied the Wigner surmise, i.e., substitution of \infty \times \infty matrices by their 2 \times 2 counterparts for the computation of level spacing distributions, to random matrix ensembles in transition between two universality classes. I examine the accuracy and the range of validity of the surmise for the crossover between the Gaussian orthogonal and unitary ensembles by contrasting them with the large-N results that I evaluated using the Nystrom-type method for the Fredholm determinant. The surmised expression at the best-fitting parameter provides a good approximation for 0 \lesssim s \lesssim 2, i.e., the validity range of the original surmise.