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

We apply universality limits to asymptotics of spacing of zeros xkn of orthogonal polynomials, for weights with compact support and for exponential weights. A typical result islimn→∞xkn-xk+1,nK˜nxkn,xkn=1under minimal hypotheses on the weight, with K˜n denoting a normalized reproducing kernel. Moreover, for exponential weights, we derive asymptotics for the differentiated kernels:Knr,sx,x=∑k=0n-1pkrxpksx.