Local limits for orthogonal polynomials for varying measures via universality

We consider orthogonal polynomials p_ne^−2nQ_n ,x for varying measures and use universality limits to prove ”local limits” limn→∞[Formula presented]e^{−[Formula presented]z}=cosπz.Here y_jn is a local maximum point of p_ne^−nQ_n in the ”bulk” of the support, K̃_ny_jn,y_jn is the normalized reproducing kernel, and the limit holds uniformly for z in compact subsets of the plane. We also consider local limits at the ”soft edge” of the spectrum, which involve the Airy function.