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Coalescence of geodesics in exactly solvable models of last passage percolation. (English) Zbl 07116349
Summary: 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 independent and identically distributed exponential weights on the vertices. Fix two points $$v_1$$= (0, 0) and $$v_2 = (0, \left\lfloor\right. k^{2/3}\left.\right\rfloor)$$ 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 $$n \gg k$$. Our object of study is the point of coalescence, i.e., the point $$v \in \Gamma_1 \cap \Gamma_2$$ with smallest $$|v|_1$$. We establish that the distance to coalescence $$|v|_1$$ scales as $$k$$, by showing the upper tail bound $$\mathbb{P}(| v |_1 > R k) \leq R^{- c}$$ 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 $$v_3 = (- \left\lfloor\right. k^{2/3}\left.\right\rfloor, \left\lfloor\right. k^{2/3}\left.\right\rfloor)$$ and $$v_4 = (\left\lfloor\right. k^{2/3}\left.\right\rfloor, - \left\lfloor\right. k^{2/3} \left.\right\rfloor)$$, we establish the optimal tail estimate $$\mathbb{P}(| v |_1 > R k) \asymp R^{- 2 / 3}$$, for the point of coalescence $$v$$. This answers a question left open by Pimentel [Ann. Probab. 44, No. 5, 3187–3206 (2016; Zbl 1361.60095)] who proved the corresponding lower bound.