Vector energy and large deviation

Abstract

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

$${E_Q}(\mu ): = \sum\limits_{i,j = 1}^d {{c_{i,j}}I({\mu _i},{\mu _j}) + 2\sum\limits_{j = 1}^d {\int_{{K_j}} {{Q_j}d{\mu _j}} } } $$

of a d-tuple of positive measures µ = (µ₁, …, 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.