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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.
©2019 American Institute of Physics

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
53C22 Geodesics in global differential geometry
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