Abstract
This chapter is based on the paper by Kozma-Nachmias [162] with some simplification by [197]. Our framework here is unimodular transitive graphs that contains \({\mathbb{Z}}^{d}\) as a typical example.
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References
M. Aizenman, D.J. Barsky, Sharpness of the phase transition in percolation models. Commun. Math. Phys. 108, 489–526 (1987)
M. Aizenman, C.M. Newman, Tree graph inequalities and critical behavior in percolation models. J. Stat. Phys. 36, 107–143 (1984)
M.T. Barlow, T. Kumagai, Random walk on the incipient infinite cluster on trees. Illinois J. Math. 50, 33–65 (2006) (electronic)
D.J. Barsky, M. Aizenman, Percolation critical exponents under the triangle condition. Ann. Probab. 19, 1520–1536 (1991)
B. Bollobás, Random Graphs, 2nd edn. (Cambridge University Press, Cambridge, 2001)
Z.-Q. Chen, P. Kim, T. Kumagai, Discrete approximation of symmetric jump processes on metric measure spaces. Probab. Theory Relat. Fields 155, 703–749 (2013)
N. Crawford, A. Sly, Simple random walks on long range percolation clusters I: heat kernel bounds. Probab. Theory Relat. Fields 154, 753–786 (2012)
N. Crawford, A. Sly, Simple random walks on long range percolation clusters II: scaling limits. Ann. Probab. 41, 445–502 (2013)
G. Grimmett, Percolation, 2nd edn. (Springer, Berlin, 1999)
T. Hara, Decay of correlations in nearest-neighbour self-avoiding walk, percolation, lattice trees and animals. Ann. Probab. 36, 530–593 (2008)
T. Hara, R. van der Hofstad, G. Slade, Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models. Ann. Probab. 31, 349–408 (2003)
M. Heydenreich, R. van der Hofstad, T. Hulshof, Random walk on the high-dimensional IIC. ArXiv:1207.7230 (2012)
R. van der Hofstad, A.A. Járai, The incipient infinite cluster for high-dimensional unoriented percolation. J. Stat. Phys. 114, 625–663 (2004)
G. Kozma, Percolation on a product of two trees. Ann. Probab. 39, 1864–1895 (2011)
G. Kozma, The triangle and the open triangle. Ann. Inst. Henri Poincaré Probab. Stat. 47, 75–79 (2011)
G. Kozma, A. Nachmias, The Alexander-Orbach conjecture holds in high dimensions. Invent. Math. 178, 635–654 (2009)
G. Kozma, A. Nachmias, Arm exponents in high dimensional percolation. J. Amer. Math. Soc. 24, 375–409 (2011)
A. Nachmias, Y. Peres, Critical random graphs: diameter and mixing time. Ann. Probab. 36, 1267–1286 (2008)
A. Sapozhnikov, Upper bound on the expected size of intrinsic ball. Electron. Commun. Probab. 15, 297–298 (2010)
R. Schonmann, Multiplicity of phase transitions and mean-field criticality on highly non-amenable graphs. Commun. Math. Phys. 219, 271–322 (2001)
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Kumagai, T. (2014). Alexander–Orbach Conjecture Holds When Two-Point Functions Behave Nicely. In: Random Walks on Disordered Media and their Scaling Limits. Lecture Notes in Mathematics(), vol 2101. Springer, Cham. https://doi.org/10.1007/978-3-319-03152-1_6
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DOI: https://doi.org/10.1007/978-3-319-03152-1_6
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