Directed last passage percolation models on the plane, where one studies the weight as well as the geometry of optimizing paths (called polymers) in a field of i.i.d. weights, are paradigm examples of models in KPZ universality class. In this article, we consider the large deviation regime, i.e., when the polymer has a much smaller (lower tail) or larger (upper tail) weight than typical. Precise asymptotics of large deviation probabilities have been obtained in a handful of the so-called exactly solvable scenarios, including the Exponential (Johansson in Commun Math Phys 209(2):437–476, 2000) and Poissonian (Deuschel and Zeitouni in Comb Probab Comput 8(03):247–263, 1999; Seppäläinen in Probab Theory Relat Fields 112(2):221–244, 1998) cases. How the geometry of the optimizing paths change under such a large deviation event was considered in Deuschel and Zeitouni (1999) where it was shown that the paths [from (0, 0) to (n, n), say] remain concentrated around the straight line joining the end points in the upper tail large deviation regime, but the corresponding question in the lower tail was left open. We establish a contrasting behaviour in the lower tail large deviation regime, showing that conditioned on the latter, in both the models, the optimizing paths are not concentrated around any deterministic curve. Our argument does not use any ingredient from integrable probability, and hence can be extended to other planar last passage percolation models under fairly mild conditions; and also to other non-integrable settings such as last passage percolation in higher dimensions.
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Observe that there is almost surely a unique maximizing path between any two vertices, by continuity of Exponential random variables.
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The authors thank Timo Seppäläinen for pointing out some relevant references, and Ofer Zeitouni for useful conversations. We also thank an anonymous referee for a careful reading of the manuscript as well as several insightful comments. Research of RB is partially supported by an ICTS-Simons Junior Faculty Fellowship and a Ramanujan Fellowship (SB/S2/RJN-097/2017) by Govt. of India. SG’s research was supported by a Miller Research Fellowship. AS is supported by NSF Grant DMS-1352013 and a Simons Investigator grant.
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Basu, R., Ganguly, S. & Sly, A. Delocalization of Polymers in Lower Tail Large Deviation. Commun. Math. Phys. 370, 781–806 (2019). https://doi.org/10.1007/s00220-019-03526-0