Mathematics Faculty Work
http://hdl.handle.net/1951/69636
2019-03-07T20:10:33ZGeodesics and Bounded Harmonic Functions on Infinite Graphs
http://hdl.handle.net/1951/69950
Geodesics and Bounded Harmonic Functions on Infinite Graphs
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.
This article has been published in the September 1991 issue of Proceedings of the American Mathematical Society.
1991-01-01T00:00:00ZCogrowth of Regular Graphs
http://hdl.handle.net/1951/69949
Cogrowth of Regular Graphs
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.
This article has been published in the September 1992 issue of Proceedings of the American Mathematical Society.
1992-01-01T00:00:00ZAmenability and superharmonic functions
http://hdl.handle.net/1951/69948
Amenability and superharmonic functions
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.
This article has been published in the October 1993 issue of Proceedings of the American Mathematical Society.
1993-01-01T00:00:00ZOn the spectrum and Martin boundary of homogeneous spaces
http://hdl.handle.net/1951/69947
On the spectrum and Martin boundary of homogeneous spaces
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.
This article has been published in the March 1995 issue of Statistics and Probability Letters.
1995-01-01T00:00:00Z