Lecture 14 – 15 : Bulk scaling limit of GUE near 0

Now we continue working with the Bulk scaling limiting, i.e., we are looking for a n and b n such that b n (Λ n − a n) converges to a " nice " limit as a point process. In the previous lecture, we have already found the Bulk scaling. From now on, we are trying to identify the limit. Recall that: • For |c| < 2, we will use the Bulk scaling σ(c) √ n(Λ n −σ(c) √ n), where σ(x) = 1 2π √ 4 − x 2 1 |x|<2 and Λ = {λ 1 , · · · , λ n }. • For β = 2 case (GUE), the Bulk scaling at 0 is 1 π √ nΛ n , and the scaled joint intensities will converges to something nice. (Here c = 0, σ(c) = 1 π). • The joint eigenvalue density of GUE is P n (λ 1 , · · · , λ n) = 1 Z n (λ) 2 e − 1 2 P λ 2 i , where (λ) = (λ i − λ j). This is unordered n-tuples live on R n Definition 1. ζ k,n (ν 1 , · · · , ν k) = P n (ν 1 , · · · , ν n)dν k+1 · · · dν n , where k is fixed, and n → ∞.