Spectral Theory of Determinantal Random Fields
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2025
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| _version_ | 1866901992463400960 |
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| author | SÉRGIO DE ANDRADE, PAULO |
| author_facet | SÉRGIO DE ANDRADE, PAULO |
| contents | Determinantal random fields, also known as determinantal point processes (DPPs), are a class of stochastic models that arise naturally in various fields, including random matrix theory, quantum mechanics, and machine learning. A key characteristic of these processes is that their correlation functions are expressed as determinants of a matrix constructed from a single function, the correlation kernel. This inherent structure endows them with strong negative correlations, or repulsion, between points. This paper provides a comprehensive exploration of the spectral theory associated with these fields. We investigate the fundamental connection between the spectral properties of the integral operator defined by the correlation kernel and the statistical behavior of the random field. Specifically, we demonstrate how the eigenvalues and eigenfunctions of this operator govern key probabilistic quantities such as point densities, gap probabilities, and number fluctuations. The analysis covers the foundational mathematical framework, reviews canonical examples arising from random matrix theory such as the sine, Airy, and Bessel kernels, and discusses the universality phenomena observed in the scaling limits. By leveraging tools from operator theory and Fredholm determinants, this work elucidates how a spectral perspective provides a unified and powerful approach to understanding the complex structure and universal laws governing determinantal random fields. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17681338 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Spectral Theory of Determinantal Random Fields SÉRGIO DE ANDRADE, PAULO Determinantal random fields, also known as determinantal point processes (DPPs), are a class of stochastic models that arise naturally in various fields, including random matrix theory, quantum mechanics, and machine learning. A key characteristic of these processes is that their correlation functions are expressed as determinants of a matrix constructed from a single function, the correlation kernel. This inherent structure endows them with strong negative correlations, or repulsion, between points. This paper provides a comprehensive exploration of the spectral theory associated with these fields. We investigate the fundamental connection between the spectral properties of the integral operator defined by the correlation kernel and the statistical behavior of the random field. Specifically, we demonstrate how the eigenvalues and eigenfunctions of this operator govern key probabilistic quantities such as point densities, gap probabilities, and number fluctuations. The analysis covers the foundational mathematical framework, reviews canonical examples arising from random matrix theory such as the sine, Airy, and Bessel kernels, and discusses the universality phenomena observed in the scaling limits. By leveraging tools from operator theory and Fredholm determinants, this work elucidates how a spectral perspective provides a unified and powerful approach to understanding the complex structure and universal laws governing determinantal random fields. |
| title | Spectral Theory of Determinantal Random Fields |
| url | https://doi.org/10.5281/zenodo.17681338 |