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February 2022 Hydrodynamic limit for a 2D interlaced particle process
Vincent Lerouvillois, Fabio Toninelli
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Ann. Appl. Probab. 32(1): 167-190 (February 2022). DOI: 10.1214/21-AAP1674


The Markov dynamics of interlaced particle arrays, introduced by A. Borodin and P. L. Ferrari in (Comm. Math. Phys. 325 (2014) 603–684), is a classical example of (2+1)-dimensional random growth model belonging to the so-called Anisotropic KPZ universality class. In (Comm. Pure Appl. Math. 72 (2018) 620–666), a hydrodynamic limit—the convergence of the height profile, after space/time rescaling, to the solution of a deterministic Hamilton–Jacobi PDE with nonconvex Hamiltonian—was proven when either the initial profile is convex, or for small times, before the solution develops shocks. In the present work, we give a simpler proof, that works for all times and for all initial profiles for which the limit equation makes sense. In particular, the convexity assumption is dropped. The main new idea is a new viewpoint about “finite speed of propagation” that allows to bypass the need of a priori control of the interface gradients, or equivalently of inter-particle distances.

Funding Statement

This work was partially funded by ANR-15-CE40-0020-03 Grant LSD.


We are grateful to Guy Barles and Vincent Calvez for help on the literature about Hamilton–Jacobi equations. We thank the referees for carefully reading the manuscript.


Download Citation

Vincent Lerouvillois. Fabio Toninelli. "Hydrodynamic limit for a 2D interlaced particle process." Ann. Appl. Probab. 32 (1) 167 - 190, February 2022.


Received: 1 March 2020; Revised: 1 December 2020; Published: February 2022
First available in Project Euclid: 27 February 2022

Digital Object Identifier: 10.1214/21-AAP1674

Primary: 60K35 , 82B20

Keywords: Hydrodynamic limit , stochastic growth processes

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.32 • No. 1 • February 2022
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