Mathematics Faculty Work

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Recent Submissions

Now showing 1 - 5 of 20
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    Geodesics and Bounded Harmonic Functions on Infinite Graphs
    (1991) Northshield, Sam
    It 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.
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    Cogrowth of Regular Graphs
    (Proceedings of the American Mathematical Society, 1992) Northshield, Sam
    Let 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.
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    Amenability and superharmonic functions
    (Proceedings of the American Mathematical Society, 1993) Northshield, Sam
    Let 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.
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    On the spectrum and Martin boundary of homogeneous spaces
    (Statistics and Probability Letters, 1995) Northshield, Sam
    Given 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.
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    On the Commute Time of Random Walks on Graphs
    (Brazilian Journal of Probability and Statistics, 1995) Northshield, Sam; Palacios, Jose Luis
    Given 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]