Large deviation for outlying coordinates in ββ ensembles

For YY a subset of the complex plane, a ββ ensemble is a sequence of probability measures Probn,β,QProbn,β,Q on YnYn for n=1,2,…n=1,2,… depending on a positive real parameter ββ and a real-valued continuous function QQ on YY. We consider the associated sequence of probability measures on YY where the probability of a subset WW of YY is given by the probability that at least one coordinate of YnYn belongs to WW. With appropriate restrictions on Y,QY,Q we prove a large deviation principle for this sequence of probability measures. This extends a result of Borot–Guionnet to subsets of the complex plane and to ββ ensembles defined with measures using a Bernstein–Markov condition.