The Eigenfunctions of the Transfer Operator for the Dyson model in a field
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918082390261760 |
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| author | Makhmudov, Mirmukhsin |
| author_facet | Makhmudov, Mirmukhsin |
| contents | The recent works \cite{EFMV2024} and \cite{JOP2023} have studied the spectral properties of the Dyson model in the absence of an external field. This paper is a continuation of \cite{EFMV2024} and aims to bridge the gap in the literature by investigating the Dyson model in a field.\\ In this paper, we prove that, for high temperatures or strong magnetic fields, there exists a non-negative, integrable (with respect to the unique half-line Gibbs measure) eigenfunction of the transfer operator for the Dyson model if $α\in(\frac 3 2,2]$. However, unlike in the zero-magnetic-field case, this eigenfunction is not continuous. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_09588 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Eigenfunctions of the Transfer Operator for the Dyson model in a field Makhmudov, Mirmukhsin Probability Mathematical Physics Dynamical Systems The recent works \cite{EFMV2024} and \cite{JOP2023} have studied the spectral properties of the Dyson model in the absence of an external field. This paper is a continuation of \cite{EFMV2024} and aims to bridge the gap in the literature by investigating the Dyson model in a field.\\ In this paper, we prove that, for high temperatures or strong magnetic fields, there exists a non-negative, integrable (with respect to the unique half-line Gibbs measure) eigenfunction of the transfer operator for the Dyson model if $α\in(\frac 3 2,2]$. However, unlike in the zero-magnetic-field case, this eigenfunction is not continuous. |
| title | The Eigenfunctions of the Transfer Operator for the Dyson model in a field |
| topic | Probability Mathematical Physics Dynamical Systems |
| url | https://arxiv.org/abs/2502.09588 |