First pair coalescence among three geodesics
Abstract
In directed last-passage percolation with i.i.d. exponential vertex weights, the coalescence time Coal(a,b) of two semi-infinite geodesics started from vertices a and b in a fixed direction has a heavy tail with (KPZ) exponent 2/3, and hence infinite expectation. We study instead the first coalescence among triples of semi-infinite geodesics in first- and last-passage percolation models (FPP/LPP) on ℤ2 that arise from stationary compatible families, such as Busemann geodesics. For directed FPP and LPP with arbitrary weights, we prove that 𝔼[min{Coal(a,a−e1),Coal(a,a−e2)}]<∞. We also consider FPP and LPP models in which antidiagonal motion is allowed, and prove that for every integer k>0, 𝔼[min{Coal(a,a−k(e1−e2)),Coal(a,a+k(e1−e2))}]<∞. These results support the heuristic prediction that adjacent coalescence times are negatively associated.
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BibTeX
@misc{krishnan2026first,
title = {{First pair coalescence among three geodesics}},
author = {Arjun Krishnan and Firas Rassoul-Agha and Timo Seppäläinen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27433},
eprint = {2610.27433},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27433},
}
RIS
TY - UNPB
TI - First pair coalescence among three geodesics
AU - Arjun Krishnan
AU - Firas Rassoul-Agha
AU - Timo Seppäläinen
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27433
UR - https://arxiv.org/abs/2610.27433
AB - In directed last-passage percolation with i.i.d. exponential vertex weights, the coalescence time Coal(a,b) of two semi-infinite geodesics started from vertices a and b in a fixed direction has a heavy tail with (KPZ) exponent 2/3, and hence infinite expectation. We study instead the first coalescence among triples of semi-infinite geodesics in first- and last-passage percolation models (FPP/LPP) on ℤ^2 that arise from stationary compatible families, such as Busemann geodesics. For directed FPP and LPP with arbitrary weights, we prove that 𝔼[min{Coal(a,a−e_1),Coal(a,a−e_2)}]<∞. We also consider FPP and LPP models in which antidiagonal motion is allowed, and prove that for every integer k>0, 𝔼[min{Coal(a,a−k(e_1−e_2)),Coal(a,a+k(e_1−e_2))}]<∞. These results support the heuristic prediction that adjacent coalescence times are negatively associated.
ER -