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Physical review. E
Let P_β^(V) (N_ I) be the probability that a N× N β-ensemble of random matrices with confining potential V(x) has N_ I eigenvalues inside an interval I=[a,b] of the real line. We introduce a general formalism, based on the Coulomb gas technique and the resolvent method, to compute analytically P_β^(V) (N_ I) for large N. We show that this probability scales for large N as P_β^(V) (N_ I)≈(-β N^2 ψ^(V)(N_ I /N)), where β is the Dyson index of the ensemble. The rate function ψ^(V)(k_ I),doi:10.1103/physreve.94.032115 pmid:27739840 fatcat:zezy4cah4jevhg44qp6cxwunz4