Vector Energy and Large Deviation

TL;DR

This paper develops discretizations of weighted vector energies for measures supported on complex plane sets, establishing a large deviation principle under certain measure conditions.

Contribution

It introduces natural discretizations of vector energies and proves a large deviation principle for these measures with strong Bernstein-Markov measures.

Findings

Discretizations W() and J() are defined for vector energies.

A large deviation principle is established for the probability measures.

Results depend on the measure  being a strong Bernstein-Markov measure.

Abstract

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 \mu=(\mu_1,...,\mu_d) where \mu_j is supported in K_j and has mass r_j. We have an L^{\infty}-type discretization W(\mu) and an L^2-type discretization J(\mu) defined using a fixed measure \nu=(\nu_1,...,\nu_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 \nu=(\nu_1,...,\nu_d) is a strong Bernstein-Markov measure.