Sum rules via large deviations

Abstract

In the theory of orthogonal polynomials, sum rules are remarkable relationships between a functional defined on a subset of all probability measures involving the reverse KullbackLeibler divergence with respect to a particular distribution and recursion coefficients related to the orthogonal polynomial construction. Killip and Simon (Killip and Simon (2003)) have given a revival interest to this subject by showing a quite surprising sum rule for measures dominating the semicircular distribution on [−2, 2]. This sum rule includes a contribution of the atomic part of the measure away from [−2, 2]. In this paper, we recover this sum rule by using probabilistic tools on random matrices. Furthermore, we obtain new (up to our knowledge) magic sum rules for the reverse Kullback-Leibler divergence with respect to the Marchenko-Pastur or Kesten-McKay distributions. As in the semicircular case, these formulas include a contribution of the atomic part appearing away from the support of the reference measure.

Large deviations and a new sum rule for spectral matrix measures of the Jacobi ensemble

“…On the one hand, as discussed in Proposition 3.2 of [18] (see also the references therein), for V in (6.17), we have F ± V = F ± J , (see (3.7) and (3.6)). That is, the rate function I V is precisely the left hand side of the sum rule in Theorem 3.1.…”

“…This distribution corresponds to a random matrix with potential V (x) = −κ 1 log(x) − κ 2 log(1 − x), (6.17) see (4.3). In the scalar case p = 1, the equilibrium measure (the minimizer of the Voiculescu entropy or the limit of Σ N ) is given by KMK(κ 1 , κ 2 ), see [18], p. 515. For this potential, the assumptions (A1), (A2) and (A3) in [21] are satisfied, with matrix equilibrium measure Σ V = Σ KMK(κ 1 ,κ 2 ) and then by Theorem 3.2 of that paper, the sequence (Σ N ) n satisfies the LDP in M p,1 (R) with speed N and good rate function…”

An Algebra Model for the Higher-Order Sum Rules

“…(4.5) where in the right hand side each term contains a factor in the form of 7) where the right hand side is a multiplication of 2d terms. With (4.3), (4.5) and (4.7), it is not hard to observe that, we can express (4.4) as a summation, where each term in this summation contains a factor like…”

“…A lot of work has been done to study higher order sum rules ( [2], [3], [5], [6], [7], [8], [10], [16]). Golinskii and Zlatǒs [8] proved that Simon's conjecture is correct under the assumption that α ∈ l 4 .…”

Universality in Random Moment Problems

“…The connection between certain random matrix ensembles and canonical moments/recursion coefficients has also been used in Gamboa et al (2016) and Gamboa et al (2017) for deriving so-called sum rules for free binomial, semicircle and Marchenko-Pastur distribution.…”