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We study invariant boundary conditions for one dimensional discrete Gaussian Markov processes, basic toy models of spatial Markov processes in statistical mechanics. More precisely, we give a decomposition of boundary objects in a non trivial basis from the study of a meromorphic matrix-valued function \(\Phi \) (inherent to the model) and its singularities. This provides a simple algorithm for the explicit computation of invariant measures. As an application, we give an “eigen” version of Szego limit theorem for matrix valued trigonometric polynomials.

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One-dimensional Discrete Gaussian Markov Processes: Harmonic Decomposition of Invariant Boundary Conditions

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  • 发表日期：

    2025年

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  • Emilien Bodiot

摘要

We study invariant boundary conditions for one dimensional discrete Gaussian Markov processes, basic toy models of spatial Markov processes in statistical mechanics. More precisely, we give a decomposition of boundary objects in a non trivial basis from the study of a meromorphic matrix-valued function \(\Phi \) (inherent to the model) and its singularities. This provides a simple algorithm for the explicit computation of invariant measures. As an application, we give an “eigen” version of Szego limit theorem for matrix valued trigonometric polynomials.