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Invasion percolation on regular trees. (English) Zbl 1145.60050

The authors study the scaling limit of the Infinite Percolation Cluster (IPC) and compare it to this of the Infinite Incipient Cluster (IIC), both on rooted trees, and provide in particular a negative answer to a recent conjecture claiming that their scaling limits should be the same. They indeed show here that, on trees, IPC does not have the already known scaling limit of the ICC and even more, that these two different limit laws were mutually singular. while they are also proved to look identical locally. IPC is also proved to be stochastically dominated by the IIC. Addictionally, the author prove scaling estimates for the Simple Random Walk on the IPC starting from the root, establishing that the IPC has a spectral 4/3 equal to this of the IIC. According to the authors, there is no reason to believe that these different scaling limits will disappear for other graphs, such as \(Z^d\), on which the already mentioned conjecture is expected to be false.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
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References:

[1] Barlow, M. T., Járai, A. A., Kumagai, T. and Slade, G. (2008). Random walk on the incipient infinite cluster for oriented percolation in high dimensions. Comm. Math. Phys. · Zbl 1144.82030
[2] Barlow, M. T. and Kumagai, T. (2006). Random walk on the incipient infinite cluster on trees. Illinois J. Math. 50 33-65. · Zbl 1110.60090
[3] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York. · Zbl 0172.21201
[4] Chayes, J. T. Chayes, L. and Durrett, R. (1987). Inhomogeneous percolation problems and incipient infinite clusters. J. Phys. A: Math. Gen. 20 1521-1530.
[5] Chayes, J. T., Chayes, L. and Newman, C. M. (1985). The stochastic geometry of invasion percolation. Comm. Math. Phys. 101 383-407. · Zbl 0596.60096
[6] Häggström, O., Peres, Y. and Schonmann, R. H. (1999). Percolation on transitive graphs as a coalescent process: Relentless merging followed by simultaneous uniqueness. In Perplexing Problems in Probability: Festschrift in Honor of Harry Kesten (M. Bramson and R. Durrett, eds.) 69-90. Birkhäuser, Basel. · Zbl 0948.60098
[7] Hara, T. and Slade, G. (1990). Mean-field critical behaviour for percolation in high dimensions. Comm. Math. Phys. 128 333-391. · Zbl 0698.60100
[8] van der Hofstad, R. (2006). Infinite canonical super-Brownian motion and scaling limits. Comm. Math. Phys 265 547-583. · Zbl 1116.60048
[9] van der Hofstad, R., den Hollander, F. and Slade, G. (2002). Construction of the incipient infinite cluster for spread-out oriented percolation above 4+1 dimensions. Comm. Math. Phys. 231 435-461. · Zbl 1013.82017
[10] van der Hofstad, R. and Slade, G. (2003). Convergence of critical oriented percolation to super-Brownian motion above 4+1 dimensions. Ann. Inst. H. Poincaré Probab. Statist. 39 413-485. · Zbl 1020.60099
[11] Járai, A. A. (2003). Invasion percolation and the incipient infinite cluster in 2 D . Comm. Math. Phys. 236 311-334. · Zbl 1041.82020
[12] Kesten, H. (1986). The incipient infinite cluster in two-dimensional percolation. Probab. Theory Related Fields 73 369-394. · Zbl 0584.60098
[13] Kesten, H. (1986). Subdiffusive behavior of random walk on a random cluster. Ann. Inst. H. Poincaré Probab. Statist. 22 425-487. · Zbl 0632.60106
[14] Newman, C. M. and Stein, D. L. (1994). Spin-glass model with dimension-dependent ground state multiplicity. Phys. Rev. Lett. 72 2286-2289.
[15] Nguyen, B. G. and Yang, W.-S. (1993). Triangle condition for oriented percolation in high dimensions. Ann. Probab. 21 1809-1844. · Zbl 0806.60097
[16] Nickel, B. and Wilkinson, D. (1983). Invasion percolation on the Cayley tree: Exact solution of a modified percolation model. Phys. Rev. Lett. 51 71-74.
[17] Wilkinson, D. and Willemsen, J. F. (1983). Invasion percolation: A new form of percolation theory. J. Phys. A: Math. Gen. 16 3365-3376.
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