Tracy-Widom Asymptotics for a River Delta Model

  • Guillaume BarraquandEmail author
  • Mark Rychnovsky
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 282)


We study an oriented first passage percolation model for the evolution of a river delta. This model is exactly solvable and occurs as the low temperature limit of the beta random walk in random environment. We analyze the asymptotics of an exact formula from [13] to show that, at any fixed positive time, the width of a river delta of length L approaches a constant times \(L^{2/3}\) with Tracy-Widom GUE fluctuations of order \(L^{4/9}\). This result can be rephrased in terms of particle systems. We introduce an exactly solvable particle system on the integer half line and show that after running the system for only finite time the particle positions have Tracy-Widom fluctuations.


KPZ universality First passage percolation Exclusion processes Tracy-Widom distribution Integrable probability 



The authors thank Ivan Corwin for many helpful discussions and for useful comments on an earlier draft of the paper. The authors thank an anonymous reviewer for detailed and helpful comments on the manuscript. G. B. was partially supported by the NSF grant DMS:1664650. M. R. was partially supported by the Fernholz Foundation’s “Summer Minerva Fellow” program, and also received summer support from Ivan Corwin’s NSF grant DMS:1811143.


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Authors and Affiliations

  1. 1.Department of MathematicsColumbia UniversityNew YorkUSA

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