Dodziuk, Jozef Difference equations, isoperimetric inequality and transience of certain random walks. (English) Zbl 0512.39001 Trans. Am. Math. Soc. 284, 787-794 (1984). Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 4 ReviewsCited in 181 Documents MathOverflow Questions: Expander mixing lemma in combinatoric expanders MSC: 39A12 Discrete version of topics in analysis 39A70 Difference operators 47B39 Linear difference operators 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 60G50 Sums of independent random variables; random walks Keywords:isoperimetric inequality; transience; difference Laplacian; arbitrary graph; maximum principle; Harnack inequality; Cheeger’s bound for the lowest eigenvalue; random walk on a graph × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Lars V. Ahlfors and Leo Sario, Riemann surfaces, Princeton Mathematical Series, No. 26, Princeton University Press, Princeton, N.J., 1960. · Zbl 0196.33801 [2] J. Cheeger, A lower bound for the lowest eigenvalue of the Laplacian, Problems in Analysis, A Symposium in honor of S. Bochner, Princeton Univ. Press, Princeton, N.J., 1970, pp. 195-199. [3] R. Courant, K. Friedrichs and H. Lewy, Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann. 100 (1928), 32-74. · JFM 54.0486.01 [4] David R. DeBaun, \?²-cohomology of noncompact surfaces, Trans. Amer. Math. Soc. 284 (1984), no. 2, 543 – 565. · Zbl 0511.58002 [5] Jozef Dodziuk, Every covering of a compact Riemann surface of genus greater than one carries a nontrivial \?² harmonic differential, Acta Math. 152 (1984), no. 1-2, 49 – 56. · Zbl 0541.30035 · doi:10.1007/BF02392190 [6] R. J. Duffin, Discrete potential theory, Duke Math. J. 20 (1953), 233 – 251. · Zbl 0051.07203 [7] John G. Kemeny, J. Laurie Snell, and Anthony W. Knapp, Denumerable Markov chains, 2nd ed., Springer-Verlag, New York-Heidelberg-Berlin, 1976. With a chapter on Markov random fields, by David Griffeath; Graduate Texts in Mathematics, No. 40. · Zbl 0348.60090 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.