Ordered random walks and the Airy line ensemble
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911482769309696 |
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| author | Denisov, Denis FitzGerald, Will Wachtel, Vitali |
| author_facet | Denisov, Denis FitzGerald, Will Wachtel, Vitali |
| contents | The Airy line ensemble is a random collection of continuous ordered paths that plays an important role within random matrix theory and the Kardar-Parisi-Zhang universality class. The aim of this paper is to prove a universality property of the Airy line ensemble. We study growing numbers of i.i.d. continuous-time random walks which are then conditioned to stay in the same order for all time using a Doob h-transform. We consider a general class of increment distributions; a sufficient condition is the existence of an exponential moment and a log-concave density. We prove that the top particles in this system converge in an edge scaling limit to the Airy line ensemble in a regime where the number of random walks is required to grow slower than a certain power (with a non-optimal exponent 3/50) of the expected number of random walk steps. Furthermore, in a similar regime we prove that the law of large numbers and fluctuations of linear statistics agree with non-intersecting Brownian motions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17827 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ordered random walks and the Airy line ensemble Denisov, Denis FitzGerald, Will Wachtel, Vitali Probability Mathematical Physics Primary 60G50, 60K35, secondary 60G40, 60F17 The Airy line ensemble is a random collection of continuous ordered paths that plays an important role within random matrix theory and the Kardar-Parisi-Zhang universality class. The aim of this paper is to prove a universality property of the Airy line ensemble. We study growing numbers of i.i.d. continuous-time random walks which are then conditioned to stay in the same order for all time using a Doob h-transform. We consider a general class of increment distributions; a sufficient condition is the existence of an exponential moment and a log-concave density. We prove that the top particles in this system converge in an edge scaling limit to the Airy line ensemble in a regime where the number of random walks is required to grow slower than a certain power (with a non-optimal exponent 3/50) of the expected number of random walk steps. Furthermore, in a similar regime we prove that the law of large numbers and fluctuations of linear statistics agree with non-intersecting Brownian motions. |
| title | Ordered random walks and the Airy line ensemble |
| topic | Probability Mathematical Physics Primary 60G50, 60K35, secondary 60G40, 60F17 |
| url | https://arxiv.org/abs/2411.17827 |