The Annals of Probability

Invasion percolation on regular trees

Omer Angel, Jesse Goodman, Frank den Hollander, and Gordon Slade

Full-text: Open access


We consider invasion percolation on a rooted regular tree. For the infinite cluster invaded from the root, we identify the scaling behavior of its r-point function for any r≥2 and of its volume both at a given height and below a given height. We find that while the power laws of the scaling are the same as for the incipient infinite cluster for ordinary percolation, the scaling functions differ. Thus, somewhat surprisingly, the two clusters behave differently; in fact, we prove that their laws are mutually singular. In addition, we derive scaling estimates for simple random walk on the cluster starting from the root. We show that the invasion percolation cluster is stochastically dominated by the incipient infinite cluster. Far above the root, the two clusters have the same law locally, but not globally.

A key ingredient in the proofs is an analysis of the forward maximal weights along the backbone of the invasion percolation cluster. These weights decay toward the critical value for ordinary percolation, but only slowly, and this slow decay causes the scaling behavior to differ from that of the incipient infinite cluster.

Article information

Ann. Probab., Volume 36, Number 2 (2008), 420-466.

First available in Project Euclid: 29 February 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35]

Invasion percolation cluster incipient infinite cluster r-point function cluster size simple random walk Poisson point process


Angel, Omer; Goodman, Jesse; den Hollander, Frank; Slade, Gordon. Invasion percolation on regular trees. Ann. Probab. 36 (2008), no. 2, 420--466. doi:10.1214/07-AOP346.

Export citation


  • 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. To appear.
  • Barlow, M. T. and Kumagai, T. (2006). Random walk on the incipient infinite cluster on trees. Illinois J. Math. 50 33–65.
  • Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • Chayes, J. T. Chayes, L. and Durrett, R. (1987). Inhomogeneous percolation problems and incipient infinite clusters. J. Phys. A: Math. Gen. 20 1521–1530.
  • Chayes, J. T., Chayes, L. and Newman, C. M. (1985). The stochastic geometry of invasion percolation. Comm. Math. Phys. 101 383–407.
  • 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.
  • Hara, T. and Slade, G. (1990). Mean-field critical behaviour for percolation in high dimensions. Comm. Math. Phys. 128 333–391.
  • van der Hofstad, R. (2006). Infinite canonical super-Brownian motion and scaling limits. Comm. Math. Phys 265 547–583.
  • 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.
  • 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.
  • Járai, A. A. (2003). Invasion percolation and the incipient infinite cluster in 2D. Comm. Math. Phys. 236 311–334.
  • Kesten, H. (1986). The incipient infinite cluster in two-dimensional percolation. Probab. Theory Related Fields 73 369–394.
  • Kesten, H. (1986). Subdiffusive behavior of random walk on a random cluster. Ann. Inst. H. Poincaré Probab. Statist. 22 425–487.
  • Newman, C. M. and Stein, D. L. (1994). Spin-glass model with dimension-dependent ground state multiplicity. Phys. Rev. Lett. 72 2286–2289.
  • Nguyen, B. G. and Yang, W.-S. (1993). Triangle condition for oriented percolation in high dimensions. Ann. Probab. 21 1809–1844.
  • 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.
  • Wilkinson, D. and Willemsen, J. F. (1983). Invasion percolation: A new form of percolation theory. J. Phys. A: Math. Gen. 16 3365–3376.