Monitoring for a Phase Transition in a Time Series of Wigner Matrices
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911042477490176 |
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| author | Dörnemann, Nina Kokoszka, Piotr Kutta, Tim Lee, Sunmin |
| author_facet | Dörnemann, Nina Kokoszka, Piotr Kutta, Tim Lee, Sunmin |
| contents | We develop methodology and theory for the detection of a phase transition in a time-series of high-dimensional random matrices. In the model we study, at each time point \( t = 1,2,\ldots \), we observe a deformed Wigner matrix \( \mathbf{M}_t \), where the unobservable deformation represents a latent signal. This signal is detectable only in the supercritical regime, and our objective is to detect the transition to this regime in real time, as new matrix--valued observations arrive. Our approach is based on a partial sum process of extremal eigenvalues of $\mathbf{M}_t$, and its theoretical analysis combines state-of-the-art tools from random-matrix-theory and Gaussian approximations. The resulting detector is self-normalized, which ensures appropriate scaling for convergence and a pivotal limit, without any additional parameter estimation. Simulations show excellent performance for varying dimensions. Applications to pollution monitoring and social interactions in primates illustrate the usefulness of our approach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_04983 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Monitoring for a Phase Transition in a Time Series of Wigner Matrices Dörnemann, Nina Kokoszka, Piotr Kutta, Tim Lee, Sunmin Statistics Theory Probability We develop methodology and theory for the detection of a phase transition in a time-series of high-dimensional random matrices. In the model we study, at each time point \( t = 1,2,\ldots \), we observe a deformed Wigner matrix \( \mathbf{M}_t \), where the unobservable deformation represents a latent signal. This signal is detectable only in the supercritical regime, and our objective is to detect the transition to this regime in real time, as new matrix--valued observations arrive. Our approach is based on a partial sum process of extremal eigenvalues of $\mathbf{M}_t$, and its theoretical analysis combines state-of-the-art tools from random-matrix-theory and Gaussian approximations. The resulting detector is self-normalized, which ensures appropriate scaling for convergence and a pivotal limit, without any additional parameter estimation. Simulations show excellent performance for varying dimensions. Applications to pollution monitoring and social interactions in primates illustrate the usefulness of our approach. |
| title | Monitoring for a Phase Transition in a Time Series of Wigner Matrices |
| topic | Statistics Theory Probability |
| url | https://arxiv.org/abs/2507.04983 |