Spectral quantization of discrete random walks on half-line, and orthogonal polynomials on the unit circle

TL;DR

The paper develops a spectral quantization framework for discrete-time random walks on the half-line by embedding Karlin–McGregor representations into orthogonal polynomials on the unit circle via Szegedy's quantization and CMV matrices. It establishes precise links between the random-walk transition probabilities and Verblunsky coefficients, showing that the classical Szegő map connects the associated OPRL on $[-1,1]$ with OPUC on the unit circle, thereby unifying stochastic and quantum dynamics through CMV theory. The authors illustrate the approach with Jacobi-polynomial-related walks, analyze constant-transition cases yielding two-periodic Verblunsky coefficients, and provide a geometric construction of the spectrum, along with an Appendix detailing periodic coefficient cases. The work generalizes the CGMV method to arbitrary Karlin–McGregor walks, offering a versatile spectral tool for quantum walk analysis with potential implications for quantum computation and integrable-system connections.

Abstract

We define quantization scheme for discrete-time random walks on the half-line consistent with Szegedy's quantization of finite Markov chains. Motivated by the Karlin and McGregor description of discrete-time random walks in terms of polynomials orthogonal with respect to a measure with support in the segment $[-1,1]$, we represent the unitary evolution operator of the quantum walk in terms of orthogonal polynomials on the unit circle. We find the relation between transition probabilities of the random walk with the Verblunsky coefficients of the corresponding polynomials of the quantum walk. We show that the both polynomials systems and their measures are connected by the classical Szegő map. Our scheme can be applied to arbitrary Karlin and McGregor random walks and generalizes the so called Cantero-Grünbaum-Moral-Velázquez method. We illustrate our approach on example of random walks related to the Jacobi polynomials. Then we study quantization of random walks with constant transition probabilities where the corresponding polynomials on the unit circle have two-periodic real Verblunsky coefficients. We present geometric construction of the spectrum of such polynomials (in the general complex case) which generalizes the known construction for the Geronimus polynomials. In the Appendix we present the explicit form, in terms of Chebyshev polynomials of the second kind, of polynomials orthogonal on the unit circle and polynomials orthogonal on the real line with coefficients of arbitrary period.

Figures (6)

• Figure 1: Discrete random walk on the half-line
• Figure 2: Spectrum of the quantum walk operator in Szegedy's quantization
• Figure 3: Szegedy's quantization of the random walk on half-line
• Figure 4: Quantum walk on the half-line splitted into coin flips described by matrix $\mathcal{M}$ and position flips given by $\mathcal{L}$
• Figure 5: Geometric construction of the spectrum of OPUC with two-periodic Verblunsky coefficients $a$ and $b$

Theorems & Definitions (55)

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• Proposition 2.1
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• Proposition 2.2
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