# Mathematics Faculty Work

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Item Geodesics and Bounded Harmonic Functions on Infinite Graphs(1991) Northshield, SamIt is shown there that an infinite connected planar graph with a uniform upper bound on vertex degree and rapidly decreasing Green's function (relative to the simple random walk) has infinitely many pairwise finitely-intersecting geodesic rays starting at each vertex. We then demonstrate the existence of nonconstant bounded harmonic functions on the graph.Item Cogrowth of Regular Graphs(Proceedings of the American Mathematical Society, 1992) Northshield, SamLet G be a d-regular graph and T the covering tree of G. We define a cogrowth constant of G in T and express it in terms of the first eigenvalue of the Laplacian on G. As a corollary, we show that the cogrowth constant is as large as possible if and only if the first eigenvalue of the Laplacian on G is zero. Grigorchuk's criterion for amenability of finitely generated groups follows.Item Amenability and superharmonic functions(Proceedings of the American Mathematical Society, 1993) Northshield, SamLet G be a countable group and u a symmetric and aperiodic probability measure on G . We show that G is amenable if and only if every positive superharmonic function is nearly constant on certain arbitrarily large subsets of G. We use this to show that if G is amenable, then the Martin boundary of G contains a fixed point. More generally, we show that G is amenable if and only if each member of a certain family of G-spaces contains a fixed point.Item On the spectrum and Martin boundary of homogeneous spaces(Statistics and Probability Letters, 1995) Northshield, SamGiven a conservative, spatially homogeneous Markov process X on an homogeneous spaces χ, we show that if the bottom of the spectrum of the generator of X is zero then the Martin boundary of contains a unique point fixed by the isometry group of χ.Item On the Commute Time of Random Walks on Graphs(Brazilian Journal of Probability and Statistics, 1995) Northshield, Sam; Palacios, Jose LuisGiven a simple random walk on an undirected connected graph, the commute time of the vertices x and y is defined as C(x,y) = ExTy+EyTx. We give a new proof, based on the optional sampling theorem for martingales, of the formula C(x,y) = 1/(Π(y)e(y,x)) in terms of the escape probability e(y,x ) (the probability that once the random walk leaves x, it hits y before it returns to x) and the stationary distribution Π(·). We use this formula for C(x,y) to show that the maximum commute time among all barbell-type graphs in N vertices is attained for the lollipop graph and its value is O[(4N3)/27]