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Sum rules via large deviations: polynomial potentials and multi-cut regime on the unit circle

Fabrice Gamboa, Jan Nagel, Alain Rouault

TL;DR

This work advances the probabilistic derivation of sum rules for orthogonal polynomials on the unit circle in the presence of polynomial potentials, including multi-cut (arced) support. By establishing large deviation principles for spectral measures and for Verblunsky coefficients, it derives an abstract gem that governs the finiteness of a spectral-entropy functional in terms of coefficient functionals, and a one-cut sum rule under mild convergence of the Verblunsky sequence. It also develops one-parameter families of potentials to analyze ungapped and gapped phases, deriving explicit sum rules and the corresponding coefficient-entropy relations for the Gross-Witten, (1,1), and (2,0) models, with clear distinctions between ungapped and gapped regimes. A key novelty is the handling of outliers and multi-arc supports on the unit circle, together with precise counterexamples showing the limits of polynomially parameterized right-hand sides in general settings. The results provide a robust framework linking large deviations, spectral measures, and Verblunsky data, with implications for random matrix models on the unit circle and their spectral-density behavior.

Abstract

Sum rules are elegant formulas that relate entropy functionals to coefficients associated with orthogonal polynomials [Sim11]. In a series of paper (see for example [GNR16], [GNR17], [BSZ18a], [BSZ18b]), interesting connections have been established between the large theory of spectral measures built on random matrices and sum rules. In this work, we extend this approach by studying sum rules within random matrix models with polynomial potentials on the unit circle, with a particular focus on cases where the equilibrium measure lacks full support.

Sum rules via large deviations: polynomial potentials and multi-cut regime on the unit circle

TL;DR

This work advances the probabilistic derivation of sum rules for orthogonal polynomials on the unit circle in the presence of polynomial potentials, including multi-cut (arced) support. By establishing large deviation principles for spectral measures and for Verblunsky coefficients, it derives an abstract gem that governs the finiteness of a spectral-entropy functional in terms of coefficient functionals, and a one-cut sum rule under mild convergence of the Verblunsky sequence. It also develops one-parameter families of potentials to analyze ungapped and gapped phases, deriving explicit sum rules and the corresponding coefficient-entropy relations for the Gross-Witten, (1,1), and (2,0) models, with clear distinctions between ungapped and gapped regimes. A key novelty is the handling of outliers and multi-arc supports on the unit circle, together with precise counterexamples showing the limits of polynomially parameterized right-hand sides in general settings. The results provide a robust framework linking large deviations, spectral measures, and Verblunsky data, with implications for random matrix models on the unit circle and their spectral-density behavior.

Abstract

Sum rules are elegant formulas that relate entropy functionals to coefficients associated with orthogonal polynomials [Sim11]. In a series of paper (see for example [GNR16], [GNR17], [BSZ18a], [BSZ18b]), interesting connections have been established between the large theory of spectral measures built on random matrices and sum rules. In this work, we extend this approach by studying sum rules within random matrix models with polynomial potentials on the unit circle, with a particular focus on cases where the equilibrium measure lacks full support.

Paper Structure

This paper contains 36 sections, 22 theorems, 227 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Notations and definitions
  3. Spectral measure and matrix representations
  4. Randomization
  5. The workshop of Breuer-Simon-Zeitouni
  6. General results
  7. The three basic examples
  8. Main results. Sum rules and gems for proper arcs
  9. One-parameter family of potentials - Ungapped case
  10. One-parameter family of potentials - Gapped phase
  11. The (1,0) case: Gross-Witten ensemble
  12. The (1,1) gapped case
  13. $\mathrm{g}< -1$
  14. The (2,0) gapped case
  15. Large deviations
  16. ...and 21 more sections

Key Result

Theorem 2.1

killipnen2004 Under $\mathbb P^{(n)}$, the distribution of the random V-coefficients $\alpha^{(n)} :=\left(\alpha^{(n)}_0,\dots , \alpha^{(n)}_{n-1}\right)$ has a density with respect to the Lebesgue measure on $\mathbb{D}^{n-1}\times \mathbb{T}$ given by

Theorems & Definitions (28)

  • Theorem 2.1
  • Lemma 2.2
  • Proposition 3.1
  • Theorem 3.2: breuersizei2018
  • Theorem 3.3: breuersizei2018
  • Theorem 3.3: breuersizei2018
  • Theorem 4.1: Abstract gem
  • Theorem 4.2
  • Corollary 4.3
  • Remark 4.4
  • ...and 18 more