Vector energy and large deviation

For d nonpolar compact sets K 1 , …, K d ⊂ ℂ, admissible weights Q 1 , …, Q d and a positive semidefinite interaction matrix C = ( c i, j ) i, j =1, …, d with no zero column, we define natural discretizations of the weighted energy of a d -tuple of positive measures µ = (µ 1 , …, µ d ) ∈ M r ( K ), where µ j is supported in K j and has mass r j . We have an L ∞ -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 { σ k } of probability measures on M r ( K ) if ν is a strong Bernstein-Markov measure.