By Emmanuel Kowalski

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We then see that X0 = 1, Xn = ξ1 · · · ξn defines a sequence (Xn ) of G-valued random variables which form a random walk on Γ with X0 = 1. To obtain a random walk with another initial distribution µ0 , pick any G-valued random variable X0 with distribution given by µ0 , and let Xn = X0 ξ1 · · · ξn . 7. 7) holds. 8. Let Γ = C(G, S) be a Cayley graph and (Xn ) a random walk on Γ starting at X0 = 1 with independent increments (ξn ), as above. For any m and n 0, 1 Xm+n is independent of Xm and distributed like Xn .

2) diam(Γ) log |Γ| 2 2 log 1 + h(Γ) v +3 where v = max val(x). x∈V The intuitive idea is the following: to “join” x to y with a short path, we look at how many elements there are at increasing distance from x and y; the definition of the expansion ratio gives a geometric lower-bound on the number of new elements when we increase the distance by one, and at some point the sets which can be reached in n steps from both sides are so big that they have to intersect, giving a distance at most 2n by the triangle inequality.

25 (Time to reach equilibrium). For the characteristic function δx of a fixed vertex, we have val(x) δx 2 = µΓ (x) = , N so that, if Γ is not bipartite, the precise statement is val(x) val(x) 1/2 n P(Xn = x) − Γ N v− for n 1. When n is “small”, this inequality is typically trivial, because the right-hand side still dominates the limiting value. A natural measure of the time when equidistribution becomes effective is the first index n when the “error” becomes comparable in size to 12 µΓ (x): for such n, the probability P(X = n = x) is at most off by a factor 2 from its limiting value.