[Paper Review] Correlation functions for determinantal processes defined by infinite block Toeplitz minors

Tomas Berggren, Maurice Duits|arXiv (Cornell University)|Jan 30, 2019

Random Matrices and Applications37 references4 citations

TL;DR

This paper establishes double integral formulas for correlation kernels of determinantal point processes defined by infinite minors of block Toeplitz matrices, using Wiener-Hopf type factorization of matrix-valued weights. The key contribution is a systematic method yielding explicit kernel formulas, providing an alternative proof for the two-periodic Aztec diamond and enabling asymptotic analysis for broader models.

ABSTRACT

We study the correlation functions for determinantal point processes defined by products of infinite minors of block Toeplitz matrices. The motivation for studying such processes comes from doubly periodically weighted tilings of planar domains, such as the two-periodic Aztec diamond. Our main results are double integral formulas for the correlation kernels. In general, the integrand is a matrix-valued function built out of a factorization of the matrix-valued weight. In concrete examples the factorization can be worked out in detail and we obtain explicit integrands. In particular, we find an alternative proof for a formula for the two-periodic Aztec diamond recently derived in \cite{DK}. We strongly believe that also in other concrete cases the double integral formulas are good starting points for asymptotic studies.

Motivation & Objective

• To develop a systematic method for deriving double integral formulas for correlation kernels in determinantal point processes defined by infinite minors of block Toeplitz matrices.
• To extend the integrable structure of scalar Toeplitz minors to the matrix-valued case, particularly for models with doubly-periodic weights.
• To provide a general framework applicable beyond the two-periodic Aztec diamond, enabling asymptotic analysis for new classes of tiling and dimer models.
• To establish a connection between the correlation kernel and matrix-valued orthogonal polynomials via Riemann-Hilbert problems.
• To demonstrate the utility of the method by re-deriving the known double integral formula for the two-periodic Aztec diamond using a new, more direct approach.

Proposed method

• Utilizes Wiener-Hopf type factorization of the matrix-valued weight function $\phi(z)$ into $\phi_+(z)$ and $\phi_-(z)$, where $\phi_+$ is analytic inside the unit circle and $\phi_-$ outside.
• Applies the Christoffel-Darboux kernel formula for matrix-valued orthogonal polynomials to express the correlation kernel in terms of the factorized weight.
• Derives a double integral representation for the kernel using contour integrals over $\gamma_0$ (around 0) and $\gamma_{0,ad/bc}$ (around $ad/bc$), with integrand involving $A(z)^m B(z)^m$ and $A(w)^{N-m'} B(w)^{-m'}$.
• Employs the identity $\left(\begin{smallmatrix}a&b\\cz&d\end{smallmatrix}\right)\left(\begin{smallmatrix}\alpha&\beta\\\gamma z&\delta\end{smallmatrix}\right) = \text{diagonal matrices} \cdot \left(\begin{smallmatrix}\alpha&\gamma\\\beta z&\delta\end{smallmatrix}\right) \cdot \text{diagonal matrices}$ to construct the factorization explicitly.
• Uses the fact that $A(z)$ and $B(z)$ commute to simplify asymptotic analysis in the $N \to \infty$ limit, allowing simultaneous diagonalization.
• Applies Theorem 4.8 on the winding number of $\det \phi(z)$ being $2N$ and non-vanishing on the unit circle to justify the factorization and kernel formula.

Experimental results

Research questions

• RQ1 Can a systematic double integral formula for the correlation kernel be derived for determinantal point processes defined by infinite minors of block Toeplitz matrices?
• RQ2 How can the Wiener-Hopf factorization of a matrix-valued weight function be constructed explicitly for such processes?
• RQ3 Can the new method reproduce known results, such as the double integral formula for the two-periodic Aztec diamond, in a simpler and more general way?
• RQ4 What conditions on the matrix weight ensure the existence of a factorization that enables explicit kernel computation?
• RQ5 To what extent can this framework be generalized to other models with doubly-periodic weights beyond the two-periodic Aztec diamond?

Key findings

• A double integral formula for the correlation kernel is derived for infinite minors of block Toeplitz matrices under the condition of a Wiener-Hopf factorization of the matrix weight.
• The kernel is expressed as a sum of a single contour integral and a double contour integral, with integrand involving matrix-valued functions $A(z)$, $B(z)$, and their powers.
• The formula provides an alternative proof for the double integral kernel of the two-periodic Aztec diamond, previously derived via inverse Kasteleyn matrix methods.
• The matrix-valued functions $A(z)$ and $B(z)$ commute, enabling simultaneous diagonalization and simplifying asymptotic analysis in the $N \to \infty$ limit.
• The method is general and applicable to other models with rational matrix weights and suitable factorization properties, such as $2\times2$-periodic lozenge tilings of hexagons.
• The existence of the factorization is guaranteed when the winding number of $\det \phi(z)$ is $2N$ and $\phi(z)$ has no zeros or poles on the unit circle.