Coalescence estimates for the corner growth model with exponential weights

Timo Seppäläinen, Xiao Shen

Published 2019-11-09Version 1

We establish estimates for the coalescence time of semi-infinite directed geodesics in the planar corner growth model with i.i.d. exponential weights. There are four estimates: upper and lower bounds for both fast and slow coalescence on the correct scale with exponent $3/2$. The lower bound for fast coalescence is new and has optimal exponential order of magnitude. For the other three we provide proofs that do not rely on integrable probability or on the connection with the totally asymmetric simple exclusion process, in order to provide a template for extension to other models. We utilize a geodesic duality introduced by Pimentel and properties of the increment-stationary last-passage percolation process.