Abstract
We prove a strong law of large numbers for directed last passage times in an independent but inhomogeneous exponential environment. Rates for the exponential random variables are obtained from a discretisation of a speed function that may be discontinuous on a locally finite set of discontinuity curves. The limiting shape is cast as a variational formula that maximises a certain functional over a set of weakly increasing curves.
On montre une loi des grands nombres pour les temps de dernier passage dirigé dans un environnement indépendant mais inhomogène et exponentiel. Les taux des variables exponentielles sont obtenues a partir d’une discretisation d’une fonction de vitesse macroscopique qui pourrait être discontinue sur un ensemble localement fini des courbes de discontinuité. La forme à la limite est déterminée par une formule des variations qui maximise une certaine fonctionnel sur un ensemble des courbes faiblement croissantes.
Funding Statement
N. Georgiou was partially supported by the EPSRC First Grant EP/P021409/1: The flat edge in last passage percolation, and by “The Dr Perry James (Jim) Browne Research Centre on Mathematics and its Applications” individual grant.
Acknowledgements
We would like to thank an anonymous referee for a careful reading of the manuscript, several comments that improved the readability, and relevant bibliography suggestions.
Citation
Federico Ciech. Nicos Georgiou. "Last passage percolation in an exponential environment with discontinuous rates." Ann. Inst. H. Poincaré Probab. Statist. 57 (4) 2165 - 2188, November 2021. https://doi.org/10.1214/21-AIHP1172
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