2017/04/18 by Riddhipratim Basu, Basu, Riddhipratim, Sourav Sarkar +3 · 1 citation
Mathematics · #Stochastic processes and statistical mechanics #Random Matrices and Applications #Markov Chains and Monte Carlo Methods
paper · pdf · doi:10.48550/arxiv.1704.05219
Coalescence of semi-infinite geodesics remains a central question in planar\nfirst passage percolation. In this paper we study finer properties of the\ncoalescence structure of finite and semi-infinite geodesics for exactly\nsolvable models of last passage percolation. Consider directed last passage\npercolation on \ℤ2 with i.i.d. exponential weights on the vertices.\nFix two points v1=(0,0) and v2=(0, lfloor k2/3 rfloor) for some\nk>0, and consider the maximal paths \Γ1 and \Γ2 starting at\nv1 and v2 respectively to the point (n,n) for n\≫ k. Our object of\nstudy is the point of coalescence, i.e., the point v\∈ \Γ1\∩ \Γ2\nwith smallest |v|1. We establish that the distance to coalescence |v|1\nscales as k, by showing the upper tail bound \ℙ(|v|1> Rk) \≤\nR-c for some c>0.\n We also consider the problem of coalescence for semi-infinite geodesics. For\nthe almost surely unique semi-infinite geodesics in the direction (1,1)\nstarting from v3=(- lfloor k2/3 rfloor , lfloor k2/3 rfloor) and\nv4=( lfloor k2/3 rfloor ,- lfloor k2/3 rfloor), we establish the\noptimal tail estimate \ℙ(|v|1> Rk) asymp R-2/3, for the point of\ncoalescence v. This answers a question left open by Pimentel (Ann. Probab.,\n2016) who proved the corresponding lower bound.\n