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2021 Flats, spikes and crevices: the evolving shape of the inhomogeneous corner growth model
Elnur Emrah, Christopher Janjigian, Timo Seppäläinen
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Electron. J. Probab. 26: 1-45 (2021). DOI: 10.1214/21-EJP595

Abstract

We study the macroscopic evolution of the growing cluster in the exactly solvable corner growth model with independent exponentially distributed waiting times. The rates of the exponentials are given by an addivitely separable function of the site coordinates. When computing the growth process (last-passage times) at each site, the horizontal and vertical additive components of the rates are allowed to also vary respectively with the column and row number of that site. This setting includes several models of interest from the literature as special cases. Our main result provides simple explicit variational formulas for the a.s. first-order asymptotics of the growth process under a decay condition on the rates. Subject to further mild conditions, we prove the existence of the limit shape and describe it explicitly. We observe that the boundary of the limit shape can develop flat segments adjacent to the axes and spikes along the axes. Furthermore, we record the formation of persistent macroscopic spikes and crevices in the cluster that are nonetheless not visible in the limit shape. As an application of the results for the growth process, we compute the flux function and limiting particle profile for the TASEP with the step initial condition and disorder in the jump rates of particles and holes. Our methodology is based on concentration bounds and estimating the boundary exit probabilities of the geodesics in the increment-stationary version of the model, with the only input from integrable probability being the distributional invariance of the last-passage times under permutations of columns and rows.

Funding Statement

E. Emrah was partially supported by the grant KAW 2015.0270 from the Knut and Alice Wallenberg Foundation and by the Mathematical Sciences Department at Carnegie Mellon University through a postdoctoral position. C. Janjigian was partially supported by National Science Foundation grant DMS-1954204 and by a postdoctoral grant from the Fondation Sciences Mathématiques de Paris while working at Université Paris Diderot. T. Seppäläinen was partially supported by National Science Foundation grants DMS-1602486 and DMS-1854619, and by the Wisconsin Alumni Research Foundation.

Acknowledgments

The authors are grateful to an anonymous referee for helpful comments.

Citation

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Elnur Emrah. Christopher Janjigian. Timo Seppäläinen. "Flats, spikes and crevices: the evolving shape of the inhomogeneous corner growth model." Electron. J. Probab. 26 1 - 45, 2021. https://doi.org/10.1214/21-EJP595

Information

Received: 15 January 2020; Accepted: 30 January 2021; Published: 2021
First available in Project Euclid: 23 March 2021

arXiv: 1908.09319
Digital Object Identifier: 10.1214/21-EJP595

Subjects:
Primary: 60K35, 60K37

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