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Random graph asymptotics on high-dimensional tori. (English) Zbl 1128.82010
In the paper it is considered Bernoulli bond percolation on the hypercubic lattice \(\mathbb Z^d\) and the finite torus \(T_{r,d} = \{-[r/2], \dots, [r/2]- 1\}^d\). The scaling of the largest critical percolation cluster on a large d-dimensional torus, for nearest-neighbor percolation in sufficiently high dimensions, or when \(d > 6\) for sufficiently spread-out percolation is investigated. Using coupling argument it is established that the largest critical cluster is, with high probability, bounded above by a large constant times \(V^{2/3}\) and below by a small constant times \(V^{2/3}(\log V)^{-4/3}\), where \(V\) is the volume of the torus. A simple criterion in terms of the subcritical percolation two-point function on \(\mathbb Z^d\) is given under which the lower bound can be improved to small constant times \(V^{2/3}\), i.e., it is proven random graph asymptotics for the largest critical cluster on the high-dimensional torus.
This establishes a conjecture by M. Aizenman [Nucl. Phys., B 485, No. 3, 551–582 (1997; Zbl 0925.82112)], apart from logarithmic corrections. Other implications of these results on the dependence on boundary conditions for high-dimensional percolation are disscused. The used method is crucially based on the results of [C. Borgs, J. T. Chayes, R. van der Hofstad, G. Slade and J. Spencer, Ann. Probab. 33, No. 5, 1886–1944 (2005; Zbl 1079.05087); R. van der Hofstad and G. Slade, Random Struct. Algorithms 27, No. 3, 331–357 (2005; Zbl 1077.60077)], where the \(V^{2/3}\) scaling was proved subject to the assumption that a suitably defined critical window contains the percolation threshold on \(\mathbb Z^d\).

MSC:
82B43 Percolation
60K35 Interacting random processes; statistical mechanics type models; percolation theory
05C80 Random graphs (graph-theoretic aspects)
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[1] Aizenman M. (1997) On the number of incipient spanning clusters. Nucl. Phys. B [FS] 485: 551–582 · Zbl 0925.82112
[2] Aizenman, M.: Private communication (2005)
[3] Aizenman M., Barsky D.J. (1987) Sharpness of the phase transition in percolation models. Commun. Math. Phys. 108: 489–526 · Zbl 0618.60098
[4] Aizenman M., Newman C.M. (1984) Tree graph inequalities and critical behavior in percolation models. J. Stat. Phys. 36: 107–143 · Zbl 0586.60096
[5] Aldous D.J. (1997) Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Probab. 25: 812–854 · Zbl 0877.60010
[6] Barsky D.J., Aizenman M. (1991) Percolation critical exponents under the triangle condition. Ann. Probab. 19: 1520–1536 · Zbl 0747.60093
[7] Benjamini I., Kozma G. (2005) Loop-erased random walk on a torus in dimension 4 and above. Commun. Math. Phys. 259: 257–286 · Zbl 1083.60007
[8] Benjamini, I., Schramm, O.: Percolation beyond \({\mathbb{Z}^d}\) , many questions and a few answers. Elect. Comm. in Probab. 1, 71–82 (1996) · Zbl 0890.60091
[9] van den Berg J., Kesten H. (1985) Inequalities with applications to percolation and reliability. J. Appl. Probab. 2: 556–569 · Zbl 0571.60019
[10] Bollobás, B.: Random Graphs. London Academic Press (1985) · Zbl 0567.05042
[11] Borgs C., Chayes J.T., van der Hofstad R., Slade G., Spencer J. (2005) Random subgraphs of finite graphs: I The scaling window under the triangle condition. Random Struct. Alg. 27: 331–357 · Zbl 1076.05071
[12] Borgs C., Chayes J.T., van der Hofstad R., Slade G., Spencer J. (2005) Random subgraphs of finite graphs: II The lace expansion and the triangle condition. Ann. Probab. 33: 1886–1944 · Zbl 1079.05087
[13] Borgs C., Chayes J.T., Kesten H., Spencer J. (1999) Uniform boundedness of critical crossing probabilities implies hyperscaling. Random Struct. Alg. 15: 368–413 · Zbl 0940.60089
[14] Borgs C., Chayes J.T., Kesten H., Spencer J. (2001) The birth of the infinite cluster: finite-size scaling in percolation. Commun. Math. Phys. 224: 153–204 · Zbl 1038.82035
[15] Erdös P., Rényi A. (1960) On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl. 5: 17–61 · Zbl 0103.16301
[16] Grimmett G. (1999) Percolation. 2nd edition Berlin, Springer · Zbl 0926.60004
[17] Hara T. (1990) Mean-field critical behaviour for correlation length for percolation in high dimensions. Prob. Th. Rel. Fields 86: 337–385 · Zbl 0685.60102
[18] Hara, T.: Decay of correlations in nearest-neighbour self-avoiding walk, percolation, lattice trees and animals. http://arxiv.org/abs/math-ph/0504021 (2005)
[19] Hara T., van der Hofstad R., Slade G. (2003) Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models. Ann. Probab. 31: 349–408 · Zbl 1044.82006
[20] Hara T., Slade G. (1990) Mean-field critical behaviour for percolation in high dimensions. Commun. Math. Phys. 128: 333–391 · Zbl 0698.60100
[21] Hara T., Slade G. (1994) Mean-field behaviour and the lace expansion. In: Grimmett G. (eds) Probability and Phase Transition. Kluwer, Dordrecht · Zbl 0831.60107
[22] Hara T., Slade G. (2000) The scaling limit of the incipient infinite cluster in high-dimensional percolation. I. Critical exponents. J. Stat. Phys. 99: 1075–1168 · Zbl 0968.82016
[23] Hara T., Slade G. (2000) The scaling limit of the incipient infinite cluster in high-dimensional percolation. II. Integrated super-Brownian excursion. Commun. Math. Phys. 128: 333–391 · Zbl 0698.60100
[24] van der Hofstad R., den Hollander F., Slade G. (2002) Construction of the incipient infinite cluster for spread-out oriented percolation above 4 + 1 dimensions. Commun. Math. Phys. 231: 435–461 · Zbl 1013.82017
[25] van der Hofstad R., Járai A.A. (2004) The incipient infinite cluster for high-dimensional unoriented percolation. J. Stat. Phys. 114: 625–663 · Zbl 1061.82016
[26] Janson S., Knuth D.E., Łuczak T., Pittel B. (1993) The birth of the giant component. Random Struct. Alg 4: 71–84 · Zbl 0766.60014
[27] Janson S., Łuczak T., Rucinski A. (2000) Random Graphs. Wiley, New York
[28] Járai A.A. (2003) Incipient infinite clusters in 2D. Ann. Probab. 31: 444–485 · Zbl 1061.60106
[29] Járai A.A. (2003) Invasion percolation and the incipient infinite cluster in 2D. Commun. Math. Phys. 236: 311–334 · Zbl 1041.82020
[30] Kesten H. (1986) The incipient infinite cluster in two-dimensional percolation. Prob. Th. Rel. Fields. 73: 369–394 · Zbl 0597.60099
[31] Kesten H. (1986) Subdiffusive behavior of random walk on a random cluster. Ann. Inst. Henri Poincaré 22: 425–487 · Zbl 0632.60106
[32] Lawler G.F. (1991) Intersections of Random Walks. Birkhäuser, Boston · Zbl 1228.60004
[33] Łuczak T., Pittel B., Wierman J.C. (1994) The structure of a random graph at the point of the phase transition. Trans. Amer. Math. Soc. 341: 721–748 · Zbl 0807.05065
[34] Menshikov M.V. (1986) Coincidence of critical points in percolation problems. So. Math. Dokl. 33: 856–859 · Zbl 0615.60096
[35] Pemantle R. (1991) Choosing a spanning tree for the integer lattice uniformly. Ann. Probab. 19: 1559–1574 · Zbl 0758.60010
[36] Peres, Y., Revelle, D.: Scaling limits of the uniform spanning tree and loop-erased random walk on finite graphs. http://arxiv.org/abs/math.PR/0410430 (to appear in Ann. Probab) (2005)
[37] Schweinsberg, J.: The loop-erased random walk and the uniform spanning tree on the four-dimensional discrete torus. http://arxiv.org/abs/math.PR/0602515 (2006) · Zbl 1183.60039
[38] Slade, G.: The Lace Expansion and its Applications. Springer Lecture Notes in Mathematics, Ecole d’Eté Probabilitès Saint-Flour, Vol. 1879, Berlin-Heidelberg-New York: Springer (2006) · Zbl 1113.60005
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