Ordered random walks and the Airy line ensemble
| Title: | Ordered random walks and the Airy line ensemble |
|---|---|
| Authors: | Denisov, Denis, FitzGerald, Will, Wachtel, Vitali |
| Publication Year: | 2024 |
| Collection: | Mathematics Mathematical Physics |
| Subject Terms: | Mathematics - Probability, Mathematical Physics, Primary 60G50, 60K35, secondary 60G40, 60F17 |
| More Details: | 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. Comment: 34 pages |
| Document Type: | Working Paper |
| Access URL: | http://arxiv.org/abs/2411.17827 |
| Accession Number: | edsarx.2411.17827 |
| Database: | arXiv |
| Description not available. |