[Paper Review] Asymptotic expansions of some Toeplitz determinants via the topological recursion

Olivier Marchal|arXiv (Cornell University)|Nov 17, 2016

Advanced Combinatorial Mathematics26 references4 citations

TL;DR

This paper provides a rigorous derivation of large-$n$ asymptotic expansions for $n\times n$ Toeplitz determinants with symbols given by indicator functions of arc-intervals on the unit circle, using a Hermitian matrix model reformulation and the Eynard-Orantin topological recursion. It establishes that coefficients in the expansion are reconstructible from spectral curves, and for a single arc, computes normalization constants via Selberg integrals, generalizing Widom's result.

ABSTRACT

In this article, we study the large $n$ asymptotic expansions of $n\ imes n$ Toeplitz determinants whose symbols are indicator functions of unions of arc-intervals of the unit circle. In particular, we use an Hermitian matrix model reformulation of the problem to provide a rigorous derivation of the general form of the large $n$ expansion when the symbol is an indicator function of either a single arc-interval or several arc-intervals with a discrete rotational symmetry. Moreover, we prove that the coefficients in the expansions can be reconstructed, up to some constants, from the Eynard-Orantin topological recursion applied to some explicit spectral curves. In addition, when the symbol is an indicator function of a single arc-interval, we provide the corresponding normalizing constants using a Selberg integral and illustrate the theoretical results with numeric simulations up to order $o\\left(\\frac{1}{n^4}\ ight)$. We also briefly discuss the situation when the number of arc-intervals increases with $n$, as well as more general Toeplitz determinants to which we may apply the present strategy.

Motivation & Objective

• To derive a complete large-$n$ expansion for Toeplitz determinants with symbols as indicator functions of arc-intervals on the unit circle.
• To establish a rigorous connection between the coefficients in the asymptotic expansion and the Eynard-Orantin topological recursion applied to explicit spectral curves.
• To resolve normalization constants for the single-arc case using Selberg integrals, extending Widom's result.
• To analyze symmetric multi-arc configurations with discrete rotational symmetry, providing expansions up to $O(1)$.
• To conjecture and numerically support the $O(1)$ term in symmetric multi-cut cases and suggest future extensions to general symbols.

Proposed method

• Reformulate the Toeplitz determinant as a Hermitian matrix model with restricted eigenvalue support.
• Derive the associated spectral curve and limiting eigenvalue density from the matrix model.
• Apply the Eynard-Orantin topological recursion to compute correlators and reconstruct asymptotic coefficients.
• Use Selberg integral identities to fix normalization constants in the single-arc case.
• Perform numerical simulations up to $o(1/n^4)$ to validate theoretical predictions.
• Analyze symmetric configurations with $d=2r+1$ or $d=2s$ intervals under rotational symmetry to derive $O(1)$ terms.

Experimental results

Research questions

• RQ1 Can the full large-$n$ asymptotic expansion of Toeplitz determinants with arc-interval symbols be rigorously derived using matrix model and topological recursion techniques?
• RQ2 How are the coefficients in the asymptotic expansion related to the Eynard-Orantin topological recursion applied to the spectral curve of the model?
• RQ3 What is the precise form of the normalization constant in the single-arc case, and can it be computed via known special functions like the Selberg integral?
• RQ4 For symmetric multi-arc configurations, can the $O(1)$ term in the expansion be reconstructed via topological recursion and conjectured based on numerical evidence?
• RQ5 Can the method be extended to general symbols on the unit circle, particularly strictly positive ones, and do sub-leading corrections still follow the topological recursion structure?

Key findings

• For a single arc-interval, the full large-$n$ expansion is rigorously derived, with normalization constants fixed via the Selberg integral, generalizing Widom's result.
• The coefficients in the asymptotic expansion are reconstructible from the Eynard-Orantin topological recursion applied to an explicit spectral curve.
• For $d=2r+1\geq3$ symmetric arc-intervals, the expansion is derived up to $O(1)$, with the $O(1)$ term depending on the remainder $m_n = n \mod (2r+1)$.
• For $d=2s\geq2$ symmetric arc-intervals, the expansion up to $O(1)$ is derived, and the $O(1)$ term is conjectured to follow a specific functional dependence on $\epsilon$, supported by numerical simulations.
• In the limit $\epsilon \to 0$, the asymptotic behavior of $Z_n(\mathcal{I}_s)$ is derived in closed form, involving products of factorials and powers of $\epsilon$, with explicit dependence on $\lfloor n/(2s) \rfloor$ and $m_n$.
• Numerical simulations confirm the theoretical predictions up to $o(1/n^4)$, validating the asymptotic expansion and the conjectured $O(1)$ term in symmetric cases.