Abstract
In this paper, we consider the geodesic tree in exponential last passage percolation. We show that for a large class of initial conditions around the origin, the line-to-point geodesic that terminates in a cylinder located around the point , and whose width and length are and , respectively, agrees in the cylinder, with the stationary geodesic sharing the same end-point. In the case of the point-to-point model where the geodesic starts from the origin, we consider width , length up to , and provide lower and upper bounds for the probability that the geodesics agree in that cylinder.
Funding Statement
O. Busani was supported by the EPSRC EP/R021449/1 Standard Grant of the UK. This study did not involve any underlying data. The work of P.L. Ferrari was partly funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy—GZ 2047/1, projekt-id 390685813 and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Projektnummer 211504053—SFB 1060.
Acknowledgments
The authors are grateful to Márton Balázs for initial discussions on the topic that led to our collaboration. We are also very grateful to an anonymous referee for careful reading and a number of constructive remarks.
Citation
Ofer Busani. Patrik L. Ferrari. "Universality of the geodesic tree in last passage percolation." Ann. Probab. 50 (1) 90 - 130, January 2022. https://doi.org/10.1214/21-AOP1530
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