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Wednesday, September 23, 2020

**Abstract:** This talk will address the same topic as Tuesday's probability seminar, but in a simplified setting: we replace the d-dimensional integer lattice with the infinite d-ary tree. In this "mean-field" case, we are able to completely resolve the question of convergence for empirical distributions along FPP geodesics. The proof, which will be presented in full, leads to two interesting threads of discussion. The first concerns the difference between lattice percolation and tree percolation; here I invite discussion of an open problem. The second thread is a corollary of our methods, namely a fact about convergence of measures which I have not seen before. The talk should be accessible to all, and I will not assume that any part of Tuesday's seminar is remembered.