Abstract
Three short results are given concerning the elliptic Harnack inequality, in the context of random walks on graphs. The first is that the elliptic Harnack inequality implies polynomial growth of the number of points in balls, and the second that the elliptic Harnack inequality is equivalent to an annulus-type Harnack inequality for Green's functions. The third result uses the lamplighter group to give a counter-example concerning the relation of coupling with the elliptic Harnack inequality. 2000 Mathematics Subject Classification 31B05 (primary), 60J35, 31C25 (secondary).
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