[Paper Review] On the Complex Asymptotics of the HCIZ and BGW Integrals

Jonathan Novak|arXiv (Cornell University)|Jun 8, 2020

Random Matrices and Applications68 references17 citations

TL;DR

This paper proves a long-standing conjecture on the large-N asymptotic behavior of the Harish-Chandra/Itzykson-Zuber (HCIZ) and Brézin-Gross-Witten (BGW) integrals over unitary groups. Using exact series expansions, stable asymptotics, and functional analysis techniques, it establishes a complete topological expansion in powers of $ N^{2-2g} $, with analytic free energies $ F_N^{(g)} $ and $ G_N^{(g)} $ that converge uniformly for small coupling $ z $ and bounded matrix spectra, confirming the existence of a strong coupling expansion with integer topological coefficients.

ABSTRACT

In this paper, we prove a longstanding conjecture on the asymptotic behavior of a pair of oscillatory matrix integrals: the Harish-Chandra/Itzykson-Zuber (HCIZ) integral, and the Brezin-Gross-Witten (BGW) integral. The main result gives a complete asymptotic expansion of these integrals for small complex parameters. The coefficients of these asymptotic expansions are generating functions for monotone Hurwitz numbers sorted by genus.

Motivation & Objective

• To resolve a longstanding conjecture on the $ N \to \infty $ asymptotics of the HCIZ and BGW integrals in random matrix theory and quantum field theory.
• To establish a complete topological expansion for the logarithms of these integrals on the scale $ N^{2-2g} $, valid uniformly for small complex $ z $ and bounded spectral radii of $ A, B $.
• To prove that the coefficients $ F_N^{(g)} $ and $ G_N^{(g)} $ are analytic functions of $ z, A, B $, with uniform bounds independent of $ N $, ensuring the asymptotic series is well-behaved.
• To confirm that the universal topological coefficients $ F_g(\alpha,\beta) $ and $ G_g(\beta) $ are integers, as predicted by topological field theory and combinatorics.

Proposed method

• Derives absolutely convergent series expansions for $ I_N $ and $ J_N $ using combinatorial identities involving the length of the longest increasing subsequence (LIS) and matrix integrals over $ \mathrm{U}(N) $.
• Introduces a stable asymptotic framework by analyzing the convergence of the series for $ \log I_N $ and $ \log J_N $ in terms of $ N $-dependent free energies $ F_N^{(g)} $ and $ G_N^{(g)} $.
• Applies complex analytic techniques, including Cauchy estimates and the Borel-Carathéodory inequality, to control the uniform boundedness of remainder terms in the asymptotic expansion.
• Uses the structure of symmetric polynomials and character expansions to express the integrals in terms of power sum symmetric polynomials $ p_\alpha $ and $ p_\beta $, enabling topological factorization.
• Establishes uniform bounds on the remainder terms $ \Delta_N^{(k)} $ via majorization and analytic continuation, proving their uniform convergence in $ N $ for fixed $ k $.
• Demonstrates that the coefficients $ F_g(\alpha,\beta) $ and $ G_g(\beta) $ are integers by showing their generating functions are analytic and their norms are uniformly bounded.

Experimental results

Research questions

• RQ1 Does the HCIZ integral admit a complete asymptotic expansion in powers of $ N^{2-2g} $ as $ N \to \infty $, with analytic coefficients depending on $ z, A, B $?
• RQ2 Are the coefficients $ F_N^{(g)} $ and $ G_N^{(g)} $ uniformly bounded in $ N $ for fixed $ g $, ensuring the asymptotic series is well-defined?
• RQ3 Do the universal topological coefficients $ F_g(\alpha,\beta) $ and $ G_g(\beta) $, which determine the free energies, have integer values?
• RQ4 Can the asymptotic expansion be uniformly controlled in the complex parameter $ z $ and matrix spectra, even when $ z $ is small and $ A, B $ are bounded?
• RQ5 Is the remainder term in the asymptotic expansion of $ \log I_N $ and $ \log J_N $ uniformly small in $ N $, with error $ o(N^{2-2k}) $?

Key findings

• The conjecture on the large-$ N $ asymptotics of the HCIZ and BGW integrals is fully proven, establishing a complete topological expansion in powers of $ N^{2-2g} $.
• The free energies $ F_N^{(g)} $ and $ G_N^{(g)} $ are analytic functions of $ z $, $ A $, and $ B $, with uniform bounds independent of $ N $, ensuring convergence of the asymptotic series.
• The remainder terms in the asymptotic expansion are uniformly bounded in $ N $, with error $ o(N^{2-2k}) $, uniformly for $ |z| \leq \varepsilon_M $ and spectral radii of $ A, B \leq M $.
• The topological coefficients $ F_g(\alpha,\beta) $ and $ G_g(\beta) $ are integers, confirming their combinatorial origin in genus-$ g $ Riemann surfaces.
• The proof relies on uniform boundedness of the remainder terms $ \Delta_N^{(k)} $, established via complex analysis and the Borel-Carathéodory inequality.
• The convergence is uniform in the complex parameter $ z $ and matrix spectra, validating the strong coupling expansion in quantum field theory and random matrix theory.