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Feb 7, 2018 at 14:11 comment added YCor Nice! A reference for $\rho_d\to 0$ is H. Kesten, Symmetric Random Walks on Groups Trans. AMS 92(2), 1959, 336-354. Namely Theorem 3 p347 computes $\rho_d=\sqrt{\frac{2-d^{-1}}{d}}\le \sqrt{\frac2d}$.
Feb 7, 2018 at 12:58 history answered RaphaelB4 CC BY-SA 3.0