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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.07345 |
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| _version_ | 1866929273489588224 |
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| author | Mine, Takuya Yoshida, Nobuo |
| author_facet | Mine, Takuya Yoshida, Nobuo |
| contents | We consider an operator $P_V=(1+V)P$ on $\ell^2(Z^d)$, where $P$ is the transition operator of a symmetric irreducible random walk, and $V$ is a ``sparse'' potential. We first characterize the essential spectra of this operator. Secondly, we prove that all the eigenfunctions which correspond to discrete spectra decay exponentially fast. Thirdly, we give a sufficient condition for this operator to have an absolute spectral gap at the right edge of the spectra. Finally, as an application of the absolute spectral gap and the exponential decay of the eigenfunctions, we prove a limit theorem for the random walk under the Gibbs measure associated to the potential $V$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_07345 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The transition operator of a random walk perturbated by sparse potentials Mine, Takuya Yoshida, Nobuo Spectral Theory We consider an operator $P_V=(1+V)P$ on $\ell^2(Z^d)$, where $P$ is the transition operator of a symmetric irreducible random walk, and $V$ is a ``sparse'' potential. We first characterize the essential spectra of this operator. Secondly, we prove that all the eigenfunctions which correspond to discrete spectra decay exponentially fast. Thirdly, we give a sufficient condition for this operator to have an absolute spectral gap at the right edge of the spectra. Finally, as an application of the absolute spectral gap and the exponential decay of the eigenfunctions, we prove a limit theorem for the random walk under the Gibbs measure associated to the potential $V$. |
| title | The transition operator of a random walk perturbated by sparse potentials |
| topic | Spectral Theory |
| url | https://arxiv.org/abs/2403.07345 |