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- W2607925462 abstract "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 $mathbb{Z}^2$ with i.i.d. exponential weights on the vertices. Fix two points $v_1=(0,0)$ and $v_2=(0, lfloor k^{2/3} rfloor)$ for some $k>0$, and consider the maximal paths $Gamma_1$ and $Gamma_2$ starting at $v_1$ and $v_2$ respectively to the point $(n,n)$ for $ngg k$. Our object of study is the point of coalescence, i.e., the point $vin Gamma_1cap Gamma_2$ with smallest $|v|_1$. We establish that the distance to coalescence $|v|_1$ scales as $k$, by showing the upper tail bound $mathbb{P}(|v|_1> Rk) leq R^{-c}$ 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 $v_3=(-lfloor k^{2/3} rfloor , lfloor k^{2/3}rfloor)$ and $v_4=(lfloor k^{2/3} rfloor ,- lfloor k^{2/3}rfloor)$, we establish the optimal tail estimate $mathbb{P}(|v|_1> Rk) asymp R^{-2/3}$, for the point of coalescence $v$. This answers a question left open by Pimentel (Ann. Probab., 2016) who proved the corresponding lower bound." @default.
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- W2607925462 date "2017-04-18" @default.
- W2607925462 modified "2023-09-23" @default.
- W2607925462 title "Coalescence of Geodesics in Exactly Solvable Models of Last Passage Percolation" @default.
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