We introduce and study a family of random processes on trees we call hipster random walks, special instances of which we heuristically connect to the min-plus binary trees introduced by Robin Pemantle and studied by Auffinger and Cable (Pemantle’s Min-Plus Binary Tree, 2017. arXiv:1709.07849 [math.PR]), and to the critical random hierarchical lattice studied by Hambly and Jordan (Adv Appl Probab 36(3):824–838, 2004. https://doi.org/10.1239/aap/1093962236 ). We prove distributional convergence for the processes, after rescaling, by showing that their evolutions can be understood as a discrete analogues of certain convection–diffusion equations, then using a combination of coupling arguments and results from the numerical analysis literature on convergence of numerical approximations of PDEs.

中文翻译：

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时髦的随机游走

我们介绍并研究了一系列我们称为 hipster 随机游走的树上的随机过程，我们启发式地将其连接到 Robin Pemantle 引入并由 Auffinger 和 Cable 研究的 min-plus 二叉树（Pemantle 的 Min-Plus Binary Tree，2017 . arXiv:1709.07849 [math.PR])，以及由 Hambly 和 Jordan 研究的临界随机分层格（Adv Appl Probab 36(3):824–838, 2004. https://doi.org/10.1239/aap/ 1093962236）。我们证明了这些过程的分布收敛性，在重新标度后，通过证明它们的演变可以被理解为某些对流-扩散方程的离散类似物，然后使用耦合参数的组合和数值分析文献中关于数值近似收敛的数值分析文献的结果偏微分方程。