TY - JOUR

T1 - Coalescence of geodesics in exactly solvable models of last passage percolation

AU - Basu, Riddhipratim

AU - Sarkar, Sourav

AU - Sly, Allan

N1 - Funding Information:
The authors are grateful to a referee for many useful comments and suggestions that led to improvements of both the editorial and technical quality of the paper. R.B. would like to thank Christopher Hoffman for asking a question which led to Theorem 2 and Xiao Shen for a careful reading of and numerous helpful comments on an earlier version of the paper. He would also like to thank Alan Hammond for explaining his work on coalescent polymer trees in Brownian LPP, Ron Peled for pointing out the related work,17 and Vladas Sidoravicius for many useful discussions. R.B. was partially supported by an AMS-Simons Travel Grant during the early phases of this project and is partially supported by an ICTS-Simons Junior Faculty Fellowship and a Ramanujan Fellowship (SB/S2/RJN-097/2017) from Govt. of India. This work was completed when S.S. was a graduate student at the Department of Statistics, UC Berkeley, partly supported by a Loéve Fellowship. A.S. is supported by NSF Grant No. DMS-1352013, a Simons Investigator grant, and a MacArthur Fellowship. Part of this research was performed during two visits of R.B. to the Princeton Mathematics department; he gratefully acknowledges the hospitality.

PY - 2019/9/1

Y1 - 2019/9/1

N2 - Coalescence of semi-infinite geodesics remains a central question in planar first passage percolation. In this paper, we study finer properties of the coalescence structure of finite and semi-infinite geodesics for exactly solvable models of last passage percolation. Consider directed last passage percolation on Z2 with independent and identically distributed exponential weights on the vertices. Fix two points v1 = (0, 0) and v2 = (0, k2/3) for some k > 0, and consider the maximal paths Γ1 and Γ2 starting at v1 and v2, respectively, to the point (n, n) for n ≫ k. Our object of study is the point of coalescence, i.e., the point v ∈ Γ1 ∩ Γ2 with smallest |v|1. We establish that the distance to coalescence |v|1 scales as k, by showing the upper tail bound P(|v|1>Rk)≤R-c for some c > 0. We also consider the problem of coalescence for semi-infinite geodesics. For the almost surely unique semi-infinite geodesics in the direction (1, 1) starting from v3 = (-k2/3, k2/3) and v4 = (k2/3, -k2/3), we establish the optimal tail estimate P(|v|1>Rk)R-2/3, for the point of coalescence v. This answers a question left open by Pimentel [Ann. Probab. 44(5), 3187-3206 (2016)] who proved the corresponding lower bound.

AB - Coalescence of semi-infinite geodesics remains a central question in planar first passage percolation. In this paper, we study finer properties of the coalescence structure of finite and semi-infinite geodesics for exactly solvable models of last passage percolation. Consider directed last passage percolation on Z2 with independent and identically distributed exponential weights on the vertices. Fix two points v1 = (0, 0) and v2 = (0, k2/3) for some k > 0, and consider the maximal paths Γ1 and Γ2 starting at v1 and v2, respectively, to the point (n, n) for n ≫ k. Our object of study is the point of coalescence, i.e., the point v ∈ Γ1 ∩ Γ2 with smallest |v|1. We establish that the distance to coalescence |v|1 scales as k, by showing the upper tail bound P(|v|1>Rk)≤R-c for some c > 0. We also consider the problem of coalescence for semi-infinite geodesics. For the almost surely unique semi-infinite geodesics in the direction (1, 1) starting from v3 = (-k2/3, k2/3) and v4 = (k2/3, -k2/3), we establish the optimal tail estimate P(|v|1>Rk)R-2/3, for the point of coalescence v. This answers a question left open by Pimentel [Ann. Probab. 44(5), 3187-3206 (2016)] who proved the corresponding lower bound.

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U2 - 10.1063/1.5093799

DO - 10.1063/1.5093799

M3 - Article

AN - SCOPUS:85072251305

VL - 60

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 9

M1 - 093301

ER -