Coalescence estimates for the corner growth model with exponential weights
Coalescence estimates for the corner growth model with exponential weights
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DOI：
10.1214/20-ejp489
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发表时间：
2019-11
期刊：
影响因子：
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通讯作者：
T. Seppalainen;X. Shen
中科院分区：
文献类型：
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作者：
T. Seppalainen;X. Shen
我们建立了具有独立同分布的平面角增长模型中半无限有向测地线合并时间的估计指数权重有四个估计：在具有指数3/2的正确空间尺度上的快速和缓慢聚结的概率的上界和下界。我们的证明利用了由Pimentel引入的测地线对偶和增量固定最后一次通过渗流过程的性质。对于快速聚结，我们的边界是新的，它们具有匹配的最佳指数数量级。对于缓慢的合并，我们重现边界证明了可积概率输入，除了我们的上限错过了最佳顺序的对数因子。
We establish estimates for the coalescence time of semi-infinite directed geodesics in the planar corner growth model with i.i.d. exponential weights. There are four estimates: upper and lower bounds on the probabilities of both fast and slow coalescence on the correct spatial scale with exponent $3/2$. Our proofs utilize a geodesic duality introduced by Pimentel and properties of the increment-stationary last-passage percolation process. For fast coalescence our bounds are new and they have matching optimal exponential order of magnitude. For slow coalescence we reproduce bounds proved earlier with integrable probability inputs, except that our upper bound misses the optimal order by a logarithmic factor.