Space-time boundaries for random walks and their application to operator algebras
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866915863815258112 |
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| author | Dor-On, Adam Gekhtman, Ilya Prudnikov, Pavel |
| author_facet | Dor-On, Adam Gekhtman, Ilya Prudnikov, Pavel |
| contents | We investigate the Martin boundary of the space-time Markov chain associated to a finitely supported random walk $(Γ, μ)$ with spectral radius $ρ$ and relate it to several classical compactifications of $Γ$. Assuming the strong ratio-limit property, we prove that the reduced ratio-limit compactification embeds naturally into the space-time Martin boundary. We introduce the $0$-Martin boundary, which governs the behaviour of $\infty$-harmonic functions, and show that the $0$-Martin kernels arise as rescaled limits of $λ$-Martin kernels as $λ\rightarrow 0$. For symmetric random walks on hyperbolic groups, the $0$-Martin boundary naturally covers the Gromov boundary, while the cover need not be injective in general. Our main structural theorem identifies the minimal space-time Martin boundary with the disjoint union of minimal $λ$-Martin boundaries over $λ\in [0, ρ^{-1}]$ with its natural pointwise topology. As an application, we show that the noncommutative Shilov boundary of the tensor algebra of the random walk $(Γ, μ)$ coincides with its Toeplitz $C^*$-algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_05967 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Space-time boundaries for random walks and their application to operator algebras Dor-On, Adam Gekhtman, Ilya Prudnikov, Pavel Probability Operator Algebras Primary: 60J50, 60G50. Secondary: 60J10, 47L55 We investigate the Martin boundary of the space-time Markov chain associated to a finitely supported random walk $(Γ, μ)$ with spectral radius $ρ$ and relate it to several classical compactifications of $Γ$. Assuming the strong ratio-limit property, we prove that the reduced ratio-limit compactification embeds naturally into the space-time Martin boundary. We introduce the $0$-Martin boundary, which governs the behaviour of $\infty$-harmonic functions, and show that the $0$-Martin kernels arise as rescaled limits of $λ$-Martin kernels as $λ\rightarrow 0$. For symmetric random walks on hyperbolic groups, the $0$-Martin boundary naturally covers the Gromov boundary, while the cover need not be injective in general. Our main structural theorem identifies the minimal space-time Martin boundary with the disjoint union of minimal $λ$-Martin boundaries over $λ\in [0, ρ^{-1}]$ with its natural pointwise topology. As an application, we show that the noncommutative Shilov boundary of the tensor algebra of the random walk $(Γ, μ)$ coincides with its Toeplitz $C^*$-algebra. |
| title | Space-time boundaries for random walks and their application to operator algebras |
| topic | Probability Operator Algebras Primary: 60J50, 60G50. Secondary: 60J10, 47L55 |
| url | https://arxiv.org/abs/2603.05967 |