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Vector Energy and Large Deviation

For d nonpolar compact sets K_1,...,K_d in the complex plane, d admissible weights Q_1,...,Q_d, and a positive semidefinite d x d interaction matrix C with no zero column, we define natural discretizations of the associated weighted vector energy of a d-tuple of positive measures μ=(μ_1,...,μ_d) where μ_j is supported in K_j and has mass r_j. We have an L^{\infty}-type discretization W(μ) and an L^2-type discretization J(μ) defined using a fixed measure ν=(ν_1,...,ν_d). This leads to a large deviation principle for a canonical sequence of probability measures on this space of d-tuples of positive measures if ν=(ν_1,...,ν_d) is a strong Bernstein-Markov measure.