References
Published online by Cambridge University Press: 16 May 2024
Summary
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
- Type
- Chapter
- Information
- Harmonic Functions and Random Walks on Groups , pp. 374 - 377Publisher: Cambridge University PressPrint publication year: 2024
References
Aldous, David, and Fill, James Allen. 2002. Reversible Markov chains and random walks on graphs. Unfinished monograph, recompiled 2014, available at www.stat.berkeley.edu/~aldous/RWG/book.html.Google Scholar
Amir, Gideon, and Kozma, Gady. 2017. Every exponential group supports a positive harmonic function. ArXiv:1711.00050.Google Scholar
Avez, André. 1972. Entropie des groupes de type fini. Comptes rendus de l’Académie des Sciences Paris Series A–B, 275, A1363–A1366.Google Scholar
Avez, André. 1976. Croissance des groupes de type fini et fonctions harmoniques. Pages 35–49 of: Théorie Ergodique. Springer.CrossRefGoogle Scholar
Bass, Hyman. 1972. The degree of polynomial growth of finitely generated nilpotent groups. Proceedings of the London Mathematical Society, 3(4), 603–614.CrossRefGoogle Scholar
Benjamini, Itai, Pemantle, Robin, and Peres, Yuval. 1998. Unpredictable paths and percolation. The Annals of Probability, 26(3), 1198–1211.CrossRefGoogle Scholar
Benjamini, Itai, Duminil-Copin, Hugo, Kozma, Gady, and Yadin, Ariel. 2015. Disorder, entropy and harmonic functions. The Annals of Probability, 43(5), 2332–2373.CrossRefGoogle Scholar
Benjamini, Itai, Duminil-Copin, Hugo, Kozma, Gady, and Yadin, Ariel. 2017. Minimal growth harmonic functions on lamplighter groups. New York Journal of Mathematics, 23, 833–858.Google Scholar
Blackwell, David. 1955. On transient Markov processes with a countable number of states and stationary transition probabilities. The Annals of Mathematical Statistics, 26, 654–658.CrossRefGoogle Scholar
Carne, Thomas Keith, and Varopoulos, Nicholas. 1985. A transmutation formula for Markov chains. Bulletin des Sciences Mathématiques, 109(4), 399–405.Google Scholar
Choquet, Gustave, and Deny, Jacques. 1960. Sur l’équation de convolution µ = µ ∗ σ. Comptes rendus de l’Académie des Sciences Paris, 250, 799–801.Google Scholar
Coulhon, Thierry, and Saloff-Coste, Laurent. 1993. Isopérimétrie pour les groupes et les variétés. Revista Matemática Iberoamericana, 9(2), 293–314.CrossRefGoogle Scholar
Cover, Thomas M, and Thomas, Joy A. 1991. Elements of Information Theory. John Wiley & Sons.Google Scholar
Derriennic, Yves, et al. 1980. Quelques applications du théoreme ergodique sous-additif. Astérisque, 74, 183–201.Google Scholar
Druţu, Cornelia, and Kapovich, Michael. 2018. Geometric Group Theory. American Mathematical Society.CrossRefGoogle Scholar
Durrett, Rick. 2019. Probability: Theory and Examples. Cambridge University Press.CrossRefGoogle Scholar
Dynkin, Evgenii Borisovich, and Malyutov, Mikhail Borisovich. 1961. Random walk on groups with a finite number of generators. Doklady Akademii Nauk, 137, 1042–1045.Google Scholar
Erschler, Anna, and Karlsson, Anders. 2010. Homomorphisms to R constructed from random walks. Annales de l’Institut Fourier, 60(6), 2095–2113.CrossRefGoogle Scholar
Frisch, Joshua, Hartman, Yair, Tamuz, Omer, and Vahidi-Ferdowsi, Pooya. 2019. Choquet–Deny groups and the infinite conjugacy class property. Annals of Mathematics, 190(1), 307–320.CrossRefGoogle Scholar
Furstenberg, Harry. 1963. A Poisson formula for semi-simple Lie groups. Annals of Mathematics, 77(2), 335–386.CrossRefGoogle Scholar
Furstenberg, Harry. 1971. Random walks and discrete subgroups of Lie groups. Advances in Probability and Related Topics, 1, 1–63.Google Scholar
Furstenberg, Harry. 1973. Boundary theory and stochastic processes on homogeneous spaces. Harmonic Analysis on Homogeneous Spaces, 26, 193–229.CrossRefGoogle Scholar
Grigorchuk, Rostislav. 1980. Burnside problem on periodic groups. Funktsional’nyi Analiz i ego Prilozheniya, 14(1), 53–54.Google Scholar
Grigorchuk, Rostislav. 1984. Degrees of growth of finitely generated groups, and the theory of invariant means. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 48(5), 939–985.Google Scholar
Grigorchuk, Rostislav. 1990. On growth in group theory. Pages 325–338 of: Proceedings of the International Congress of Mathematicians, vol. 1. Springer.Google Scholar
Gromov, Michael. 1981. Groups of polynomial growth and expanding maps (with an appendix by Jacques Tits). Publications Mathématiques de l’IHÉS, 53, 53–78.CrossRefGoogle Scholar
Guivarc’h, Yves. 1973. Croissance polynomiale et périodes des fonctions harmoniques. Bulletin de la Société Mathématique de France, 101, 333–379.CrossRefGoogle Scholar
Kaimanovich, Vadim A., and Vershik, Anatoly M. 1983. Random walks on discrete groups: boundary and entropy. The Annals of Probability, 11(3), 457–490.CrossRefGoogle Scholar
Karlsson, Anders, and Ledrappier, François. 2007. Linear drift and Poisson boundary for random walks. Pure and Applied Mathematics Quarterly, 3(4), 1027–1036.CrossRefGoogle Scholar
Kesten, Harry. 1959. Full Banach mean values on countable groups. Mathematica Scandinavica, 146–156.CrossRefGoogle Scholar
Kleiner, Bruce. 2010. A new proof of Gromov’s theorem on groups of polynomial growth. Journal of the American Mathematical Society, 23(3), 815–829.CrossRefGoogle Scholar
Korevaari, Nicholas J., and Schoen, Richard M. 1997. Global existence theorems for harmonic maps to non-locally compact spaces. Communications in Analysis and Geometry, 5(2), 333–387.CrossRefGoogle Scholar
Lee, James R., and Peres, Yuval. 2013. Harmonic maps on amenable groups and a diffusive lower bound for random walks. The Annals of Probability, 41(5), 3392–3419.CrossRefGoogle Scholar
Lyons, Russell. 1995. Random walks and the growth of groups. Comptes rendus de l’Académie des sciences. Série 1, Mathématique, 320(11), 1361–1366.Google Scholar
Lyons, Russell, and Peres, Yuval. 2016. Probability on Trees and Networks. Cambridge University Press. Available at https://rdlyons.pages.iu.edu/.CrossRefGoogle Scholar
Lyons, Russell, Peres, Yuval, Sun, Xin, and Zheng, Tianyi. 2017. Occupation measure of random walks and wired spanning forests in balls of Cayley graphs. ArXiv:1705.03576.Google Scholar
Meyerovitch, Tom, and Yadin, Ariel. 2016. Harmonic functions of linear growth on solvable groups. Israel Journal of Mathematics, 216(1), 149–180.CrossRefGoogle Scholar
Meyerovitch, Tom, Perl, Idan, Tointon, Matthew, and Yadin, Ariel. 2017. Polynomials and harmonic functions on discrete groups. Transactions of the American Mathematical Society, 369(3), 2205–2229.CrossRefGoogle Scholar
Milnor, John. 1968a. Growth of finitely generated solvable groups. Journal of Differential Geometry, 2(4), 447–449.CrossRefGoogle Scholar
Mok, Ngaiming. 1995. Harmonic forms with values in locally constant Hilbert bundles. Pages 433–453 of: Journal of Fourier Analysis and Applications. CRC Press.Google Scholar
Ozawa, Narutaka. 2018. A functional analysis proof of Gromov’s polynomial growth theorem. Annales scientifiques de l’École normale supérieure, 51(3), 549–556.CrossRefGoogle Scholar
Perl, Idan, and Yadin, Ariel. 2023. Polynomially growing harmonic functions on connected groups. Groups, Geometry, and Dynamics, DOI 10.4171/GGD/660.Google Scholar
Peyre, Rémi. 2008. A probabilistic approach to Carne’s bound. Potential Analysis, 29(1), 17–36.CrossRefGoogle Scholar
Pólya, Georg. 1921. Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz. Mathematische Annalen, 84(1), 149–160.CrossRefGoogle Scholar
Raoufi, Aran, and Yadin, Ariel. 2017. Indicable groups and pc < 1. Electronic Communications in Probability, 22, 1–10.CrossRefGoogle Scholar
Rosenblatt, Joseph. 1981. Ergodic and mixing random walks on locally compact groups. Mathematische Annalen, 257(1), 31–42.CrossRefGoogle Scholar
Shalom, Yehuda, and Tao, Terence. 2010. A finitary version of Gromov’s polynomial growth theorem. Geometric and Functional Analysis, 20(6), 1502–1547.CrossRefGoogle Scholar
Tits, Jacques. 1972. Free subgroups in linear groups. Journal of Algebra, 20(2), 250–270.CrossRefGoogle Scholar
Tointon, Matthew C.H. 2016. Characterisations of algebraic properties of groups in terms of harmonic functions. Groups, Geometry, and Dynamics, 10(3), 1007–1049.CrossRefGoogle Scholar
Varopoulos, Nicholas. 1985. Long range estimates for Markov chains. Bulletin des Sciences Mathématiques, 109(3), 225–252.Google Scholar
Vershik, Anatolii Moiseevich, and Kaimanovich, Vadim Adol’fovich. 1979. Random walks on groups: Boundary, entropy, uniform distribution. Doklady Akademii Nauk, 249, 15–18.Google Scholar
Woess, Wolfgang. 2000. Random Walks on Infinite Graphs and Groups. Cambridge University Press.CrossRefGoogle Scholar
Wolf, Joseph A. 1968. Growth of finitely generated solvable groups and curvature of Riemannian manifolds. Journal of Differential Geometry, 2(4), 421–446.CrossRefGoogle Scholar