Theorem: GUE R_2(s) limits R_2(0) = 0 (strong repulsion), R_2(s->inf) = 1 (decorrelation)

Theorem (GUE sine-kernel limit canonical): the GUE 2-level correlation R_2(s) = 1 - (sin(pi s)/(pi s))^2 satisfies the two universal limits: R_2(0) = 0 (beta=2 strong level repulsion - zero density-of-pairs at coincidence s=0) and R_2(s ->…