Lectures 20 – 22 : Scaling limit of β - ensembles

1 Edelman-Sutton Conjecture (cont.) 1.1 Existence of eigenvalue-eigenfunction pair for the smallest eigenvalue of cH β In the last lecture, we proved ∃{f n } and a nonnegative random variable K with ||f n || 2 2 = 1, ||f n || 2 * K a.s. ∀n, so that f n , H β f n → Λ 0 = inf ||f || 2 2 =1 f ∈L * f, H β f a.s. Also, we proved the lemma: If {f n }is bounded in L * , then we can find a subsequence {f n k } so that 1) f n k → f in L 2 2) f n k → f weak convergence in L 2 3) f n k → f uniformly on compacts 4) f n k → f weakly in L * Without loss of generality, we may assume {f n } itself has the above properties. We will show that Λ 0 = f , H β f a.s. and that (Λ 0 , f) is an eigenvalue-eigenfunction pair with the smallest eigenvalue. Proof. In the previous lecture, we proved ∃ a nonnegative random variable C(B) : sup x max(| ¯ B (x)|, |B(x) − ¯ B(x)|) log(2 + x) < C(B) < ∞. Hence, ∀ > 0, ∃ a random variable X, so that | ¯ B (x)||(1+x) and |B(x)− ¯ B(x)|| √ 1 + x, ∀x > X.