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7 - Group-Walk Random Graphs

Published online by Cambridge University Press:  20 July 2017

Agelos Georgakopoulos
Affiliation:
Mathematics Institute, University of Warwick, CV4 7AL, United Kingdom
Tullio Ceccherini-Silberstein
Affiliation:
Università degli Studi del Sannio, Italy
Maura Salvatori
Affiliation:
Università degli Studi di Milano
Ecaterina Sava-Huss
Affiliation:
Cornell University, New York
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Publisher: Cambridge University Press
Print publication year: 2017

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References

[1] A., Ancona. Negatively curved manifolds, elliptic operators, and the Martin boundary. The Annals of Mathematics, 125(3):495, 1987.Google Scholar
[2] A., Ancona. Positive harmonic functions and hyperbolicity. In Potential Theory Surveys and Problems, vol. 1344 of Lecture Notes in Mathematics, pp. 1–23. 1988.Google Scholar
[3] I., Benjamini and O., Schramm. Recurrence of distributional limits of finite planar graphs. Electronic Journal of Probability, 6, 2001.Google Scholar
[4] C., Borgs, J.T., Chayes, H., Cohn, and Y., Zhao. An Lp theory of sparse graph convergence I: limits, sparse random graph models, and power law distributions. Preprint 2014.
[5] C., Borgs, J.T., Chayes, L., Lovász, V.T., Sós, and K., Vesztergombi. Convergent sequences of dense graphs I: Subgraph frequencies, metric properties and testing. Advances in Mathematics, 6(6):1801–51, 2008.Google Scholar
[6] M., Brelot and G., Choquet. Espaces et lignes de Green. Ann. Inst. Fourier, 3:119–263, 1951.Google Scholar
[7] J.L., Doob. Boundary properties of functions with finite Dirichlet integrals. Ann. Inst. Fourier, 12:573–621, 1962.Google Scholar
[8] Jesse, Douglas. Solution of the problem of Plateau. Trans. Am. Math. Soc., 1(1):263–321, 1931.Google Scholar
[9] P.G., Doyle and J.L., Snell. Random Walks and Electrical Networks. Carus Mathematical Monographs 22, Mathematical Association of America, 1984.
[10] P., Erdös and A., Rényi. On random graphs I. Publ. Math. Debrecen, 6:290–7, 1959.Google Scholar
[11] A., Erschler. Poisson–Furstenberg boundaries, large-scale geometry and growth of groups. In Proceedings of the ICM, pp. 681–704, 2010.Google Scholar
[12] H., Furstenberg. A Poisson formula for semi-simple Lie groups. The Annals of Mathematics, 2(2):335–86, 1963.Google Scholar
[13] A., Georgakopoulos. The boundary of a square tiling of a graph coincides with the poisson boundary. To appear in Invent. Math., DOI 10.1007/s00222-015-0601-0.
[14] A., Georgakopoulos. Electrical networks from the mathematical viewpoint. Lecture notes. In preparation.
[15] A., Georgakopoulos and V., Kaimanovich. In preparation.
[16] R. Van Der, Hofstad. Random graphs and complex networks. Lecture Notes, 2013.
[17] V., Kaimanovich. The poisson formula for groups with hyperbolic properties. The Annals of Mathematics, 152(3): 659–92, 2000.Google Scholar
[18] V.A., Kaimanovich and A.M., Vershik. Random walks on discrete groups: boundary and entropy. The Annals of Probability, 3(3):457–90, 1983.Google Scholar
[19] C., Midgley. Random graphs from groups, 2014. Undergraduate research project. University of Warwick.
[20] L., Naôm. Sur le rôle de la frontière de R.S. Martin dans la théorie du potentiel. Annales Inst. Fourier, 7:183–281, 1957.Google Scholar
[21] C.M., Newman and L.S., Schulman. One-dimensional 1/|ji |s percolation models: the existence of a transition for s ≤ 2. Commun. Math. Phys., 104:547–71, 1986.Google Scholar
[22] M., Penrose. Random Geometric Graphs. Oxford University Press, 2003.
[23] A.-S., Sznitman. Vacant set of random interlacements and percolation. Annals of Mathematics, 3(3):2039–2087, 2010.Google Scholar
[24] A., Teixeira. Interlacement percolation on transient weighted graphs. 14:1604–27, 2009.
[25] C., Thomassen. Resistances and currents in infinite electrical networks. J. Combin. Theory (Series B), 49:87–102, 1990.Google Scholar
[26] Wolfgang, Woess. Denumerable Markov Chains. Generating Functions, Boundary Theory, Random Walks on Trees. EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zürich, 2009.
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