The extreme statistics of some noncolliding Brownian processes
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866914465805500416 |
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| author | Rahman, Mustazee |
| author_facet | Rahman, Mustazee |
| contents | We consider certain noncolliding interacting particle systems driven by Brownian noise. A key example is drifted Brownian motions conditioned not to intersect and related models of eigenvalues of Hermitian random matrices. We establish limit theorems for the extremal particle. We find: (i) the scaling limit of the largest eigenvalue of Brownian motion over Hermitian, positive-definite matrices, (ii) Airy process limit for the largest eigenvalue of Dyson's Brownian motion for GUE started from generic initial conditions, and (iii) a Fredholm determinant formula for the maximum of the top path among noncolliding Brownian bridges and, as a byproduct, a new formula for the law of largest eigenvalue in a particular Laguerre Orthogonal Ensemble as well as for a related point-to-line last passage percolation model. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_03206 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The extreme statistics of some noncolliding Brownian processes Rahman, Mustazee Probability Mathematical Physics We consider certain noncolliding interacting particle systems driven by Brownian noise. A key example is drifted Brownian motions conditioned not to intersect and related models of eigenvalues of Hermitian random matrices. We establish limit theorems for the extremal particle. We find: (i) the scaling limit of the largest eigenvalue of Brownian motion over Hermitian, positive-definite matrices, (ii) Airy process limit for the largest eigenvalue of Dyson's Brownian motion for GUE started from generic initial conditions, and (iii) a Fredholm determinant formula for the maximum of the top path among noncolliding Brownian bridges and, as a byproduct, a new formula for the law of largest eigenvalue in a particular Laguerre Orthogonal Ensemble as well as for a related point-to-line last passage percolation model. |
| title | The extreme statistics of some noncolliding Brownian processes |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2604.03206 |