  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 (2017; arXiv:1709.07849), and to the critical random hierarchical lattice
studied by Hambly and Jordan (2004). We prove distributional convergence for
the processes 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.
