Optimal exponent for coalescence of finite geodesics in exponential last passage percolation

Abstract

In this note, we study the model of directed last passage percolation on Z 2 , with i.i.d. exponential weight. We consider the maximum directed paths from vertices (0, k 2/3 ) and ( k 2/3 , 0) to (n, n), respectively. For the coalescence point of these paths, we show that the probability for it being > Rk far away from the origin is in the order of R −2/3 . This is motivated by a recent work of Basu, Sarkar, and Sly [7], where the same estimate was obtained for semi-infinite geodesics, and the optimal exponent for the finite case was left open. Our arguments also apply to other exactly solvable models of last passage percolation.

Local stationarity in exponential last-passage percolation

“…Theorem 2.6 shows that p c converges weakly to p ∞ c . In this theorem we also show how our stabilization result gives an alternative route from the bounds on p ∞ c given in [3] to the tail decay of | p c | earlier derived in [36].…”

“…The exponent c in (2.10) was not identified but was conjectured to be 2 /3. This was recently settled by Zhang in [36]. The theorem below shows how our stabilization result transfers the bounds (2.9) from p ∞ c to p c .…”

“…Without relying on integrable probability methods, they reproved the results of [3] up to a logarithmic error and obtained new upper and lower bounds of matching order on abnormally fast coalescence. In [36] Zhang proved the optimal bounds of − 2 /3 for coalescence of two point-to-point geodesics from two points at distance k. The proof relies on diffusive concentration of geodesic fluctuations coming from integrable probability.…”

Disjoint optimizers and the directed landscape

“…More recently, geodesic coalescence and disjointness have been studied in the more tractable context of integrable last passage percolation by using probabilistic and geometric techniques, e.g. see Hammond [Ham20]; Pimentel [Pim16]; Basu, Sarkar, Sly and Zhang [BSS19,Zha20]; Balázs, Busani, Georgiou, Rassoul-Agha, Seppäläinen, Shen [GRAS17, SS20, BBS20]. Questions of geodesic coalescence and disjointness still make sense in the directed landscape, and studying these reveals interesting probabilistic structures, e.g.…”

Mixing times for the TASEP in the maximal current phase

“…However, note that many of these exact expressions for the TASEP tend to fail under minor changes of the setup. This led to a series of works, including [7,8,9,10,11,48,59], where for example the effect of changing one rate in the TASEP, known as the slow bond problem, or the correlation of last-passage times and the coalescence of geodesics are studied. The articles have in common that they use results from integrable probability and combine them with probabilistic concepts; see also [4,55] for similar results avoiding the connection to integrable probability entirely.…”