Non-intersecting Brownian Motions and Gaussian Multiplicative Chaos
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913993357000704 |
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| author | Keles, Ahmet |
| author_facet | Keles, Ahmet |
| contents | We obtain Fisher-Hartwig asymptotics with root and jump type singularities in space-time under the law of the stationary Hermitian Ornstein-Uhlenbeck process, which serve as a dynamical generalization of earlier static results obtained by Riemann-Hilbert methods. This extends previous asymptotics by [Krasovsky 2007], [Its, Krasovsky 2008], and [Charlier 2019]. As a consequence, fractional powers of the absolute value of the characteristic polynomial of this process (and the exponential eigenvalues counting process) converge to a two dimensional Gaussian multiplicative chaos measure on an infinite strip in the subcritical phase. The dynamical Fisher-Hartwig asymptotics also provide the leading order of the log-characteristic polynomial, together with optimal bulk rigidity for non-intersecting Brownian motions. These results offer (i) the second connection between random matrix theory and Liouville quantum gravity measures after [Bourgade, Falconet 2025], by proving a dynamical generalization of the single-time convergence to the GMC from [Berestycki, Webb, Wong 2018], (ii) a dynamical extension of the maximum of the log-characteristic polynomial [Lambert, Paquette 2019] and the optimal rigidity [Claeys, Fahs, Lambert, Webb 2021]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_11505 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-intersecting Brownian Motions and Gaussian Multiplicative Chaos Keles, Ahmet Probability Mathematical Physics We obtain Fisher-Hartwig asymptotics with root and jump type singularities in space-time under the law of the stationary Hermitian Ornstein-Uhlenbeck process, which serve as a dynamical generalization of earlier static results obtained by Riemann-Hilbert methods. This extends previous asymptotics by [Krasovsky 2007], [Its, Krasovsky 2008], and [Charlier 2019]. As a consequence, fractional powers of the absolute value of the characteristic polynomial of this process (and the exponential eigenvalues counting process) converge to a two dimensional Gaussian multiplicative chaos measure on an infinite strip in the subcritical phase. The dynamical Fisher-Hartwig asymptotics also provide the leading order of the log-characteristic polynomial, together with optimal bulk rigidity for non-intersecting Brownian motions. These results offer (i) the second connection between random matrix theory and Liouville quantum gravity measures after [Bourgade, Falconet 2025], by proving a dynamical generalization of the single-time convergence to the GMC from [Berestycki, Webb, Wong 2018], (ii) a dynamical extension of the maximum of the log-characteristic polynomial [Lambert, Paquette 2019] and the optimal rigidity [Claeys, Fahs, Lambert, Webb 2021]. |
| title | Non-intersecting Brownian Motions and Gaussian Multiplicative Chaos |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2508.11505 |