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Ordered random walks and the Airy line ensemble

Denis Denisov et al.

Probability Theory and Related Fields2026https://doi.org/10.1007/s00440-026-01481-1article
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Abstract

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.

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https://doi.org/https://doi.org/10.1007/s00440-026-01481-1

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@article{denis2026,
  title        = {{Ordered random walks and the Airy line ensemble}},
  author       = {Denis Denisov et al.},
  journal      = {Probability Theory and Related Fields},
  year         = {2026},
  doi          = {https://doi.org/https://doi.org/10.1007/s00440-026-01481-1},
}

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Balanced mode · F 0.40 / M 0.15 / V 0.05 / R 0.40

F · citation impact0.50 × 0.4 = 0.20
M · momentum0.50 × 0.15 = 0.07
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R · text relevance †0.50 × 0.4 = 0.20

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