Applications of universality limits to zeros and reproducing kernels of orthogonal polynomials

Abstract

We apply universality limits to asymptotics of spacing of zeros {x kn } of orthogonal polynomials, for weights with compact support and for exponential weights. A typical result is lim n→∞ x kn − x k+1,n K n (x kn , x kn ) = 1 under minimal hypotheses on the weight, withK n denoting a normalized reproducing kernel. Moreover, for exponential weights, we derive asymptotics for the differentiated kernels:

Two extensions of Lubinsky’s universality theorem

“…Given Theorem 1.4, this is essentially a straightforward translation of [19] and [18]. We note that earlier Freud [10] had noted that the universality result, (1.16), (which he only had under very restrictive assumptions) implies clock behavior of zeros.…”

“…and, uniformly for x 0 ∈ [a, b], |α|, |β| ≤ A, one has (1.16), and so (as in Section 4 following [18]), clock behavior for the zeros.…”

Universality limits in the bulk for arbitrary measures on compact sets

“…Together with the method of [15], Levin's fundamental idea lead to universality results for …xed exponential weights on the real line. This circle of methods has also been exploited for orthogonal polynomials on the unit circle, and for universality at the edge of the spectrum [12], [13], [16], [17].…”

Universality Limits of a Reproducing Kernel for a Half-Line Schrödinger Operator and Clock Behavior of Eigenvalues

“…Like [Sim08b], [LL08], we can now deduce clock spacing of the zeros for a perturbed periodic potential. Here we prove Corollary 1.3.3.…”