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References

  • Aldous, David. Asymptotic fringe distributions for general families of random trees. Ann. Appl. Probab. 1 (1991), no. 2, 228--266. MR1102319
  • S. Alexander, R. Orbach: `Density of states on fractals: ''fractons'' ', J. Physique (Paris) Lett. 43, 625-631, (1982).
  • T. Antunovic, I. Veselic: `Spectral asymptotics of percolation hamiltonians on amenable Cayley graphs', In: Proceedings of OTAMP 2006. Operator Th.: Adv. Appl. (2007).
  • Bandyopadhyay, Antar; Steif, Jeffrey; Timár, Ádám. On the cluster size distribution for percolation on some general graphs. Rev. Mat. Iberoam. 26 (2010), no. 2, 529--550. MR2677006
  • Barlow, Martin T.; Kumagai, Takashi. Random walk on the incipient infinite cluster on trees. Illinois J. Math. 50 (2006), no. 1-4, ISBN: 0-9746986-1-X 33--65 (electronic). MR2247823
  • Batagelj, Vladimir; Pisanski, Tomaž. Hamiltonian cycles in the Cartesian product of a tree and a cycle. Discrete Math. 38 (1982), no. 2-3, 311--312. MR0676545
  • Benjamini, Itai; Lyons, Russell; Peres, Yuval; Schramm, Oded. Critical percolation on any nonamenable group has no infinite clusters. Ann. Probab. 27 (1999), no. 3, 1347--1356. MR1733151
  • A.G. Boshier: `Enlarging properties of graphs', Ph.D. thesis, Royal Holloway and Bedford New College, University of London, (1987).
  • Dimakopoulos, Vassilios V.; Palios, Leonidas; Poulakidas, Athanasios S. On the Hamiltonicity of the Cartesian product. Inform. Process. Lett. 96 (2005), no. 2, 49--53. MR2166269
  • Fisher, Michael E.; Essam, John W. Some cluster size and percolation problems. J. Mathematical Phys. 2 1961 609--619. MR0126305
  • Fontes, L. R. G.; Mathieu, P. On symmetric random walks with random conductances on ${\Bbb Z}^ d$. Probab. Theory Related Fields 134 (2006), no. 4, 565--602. MR2214905
  • Grimmett, Geoffrey. Percolation. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 321. Springer-Verlag, Berlin, 1999. xiv+444 pp. ISBN: 3-540-64902-6 MR1707339
  • Grimmett, G. R. On the number of clusters in the percolation model. J. London Math. Soc. (2) 13 (1976), no. 2, 346--350. MR0408045
  • van den Heuvel, J. Hamilton cycles and eigenvalues of graphs. Linear Algebra Appl. 226/228 (1995), 723--730. MR1344594
  • Hughes, Barry D. Random walks and random environments. Vol. 1. Random walks. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. xxii+631 pp. ISBN: 0-19-853788-3 MR1341369
  • Kesten, Harry. The critical probability of bond percolation on the square lattice equals ${1\over 2}$. Comm. Math. Phys. 74 (1980), no. 1, 41--59. MR0575895
  • Kesten, Harry. Subdiffusive behavior of random walk on a random cluster. Ann. Inst. H. Poincaré Probab. Statist. 22 (1986), no. 4, 425--487. MR0871905
  • Kesten, Harry. Scaling relations for $2$D-percolation. Comm. Math. Phys. 109 (1987), no. 1, 109--156. MR0879034
  • Kirsch, Werner; Müller, Peter. Spectral properties of the Laplacian on bond-percolation graphs. Math. Z. 252 (2006), no. 4, 899--916. MR2206633
  • Kolchin, Valentin F. Random mappings. Translated from the Russian. With a foreword by S. R. S. Varadhan. Translation Series in Mathematics and Engineering. Optimization Software, Inc., Publications Division, New York, 1986. xiv + 207 pp. ISBN: 0-911575-16-2 MR0865130
  • Kozma, Gady; Nachmias, Asaf. The Alexander-Orbach conjecture holds in high dimensions. Invent. Math. 178 (2009), no. 3, 635--654. MR2551766
  • R. Lyons, Y. Peres: `Probability on trees and networks', web-book: http://mypage.iu.edu/simrdlyons/prbtree/prbtree.html.
  • Mohar, Bojan. Isoperimetric numbers of graphs. J. Combin. Theory Ser. B 47 (1989), no. 3, 274--291. MR1026065
  • Müller, Peter; Stollmann, Peter. Spectral asymptotics of the Laplacian on supercritical bond-percolation graphs. J. Funct. Anal. 252 (2007), no. 1, 233--246. MR2357356
  • Norris, J. R. Markov chains. Reprint of 1997 original. Cambridge Series in Statistical and Probabilistic Mathematics, 2. Cambridge University Press, Cambridge, 1998. xvi+237 pp. ISBN: 0-521-48181-3 MR1600720
  • Saloff-Coste, Laurent. Lectures on finite Markov chains. Lectures on probability theory and statistics (Saint-Flour, 1996), 301--413, Lecture Notes in Math., 1665, Springer, Berlin, 1997. MR1490046
  • Sobieczky, Florian. An interlacing technique for spectra of random walks and its application to finite percolation clusters. J. Theoret. Probab. 23 (2010), no. 3, 639--670. MR2679951
  • Woess, Wolfgang. Random walks on infinite graphs and groups. Cambridge Tracts in Mathematics, 138. Cambridge University Press, Cambridge, 2000. xii+334 pp. ISBN: 0-521-55292-3 MR1743100


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