Scaling limit of the invasion percolation cluster on a regular tree

Invasion percolation on regular trees

We consider invasion percolation on a rooted regular tree. For the infinite cluster invaded from the root, we identify the scaling behaviour of its r-point function for any r ≥ 2 and of its volume both at a given height and below a given height. In addition, we derive scaling estimates for simple random walk on the cluster starting from the root. We find that while the power laws of the scaling...

Diffusion on the scaling limit of the critical percolation cluster in the diamond hierarchical lattice

We construct critical percolation clusters on the diamond hierarchical lattice and show that the scaling limit is a graph directed random recursive fractal. A Dirichlet form can be constructed on the limit set and we consider the properties of the associated Laplace operator and diffusion process. In particular we contrast and compare the behaviour of the high frequency asymptotics of the spect...

Invasion Percolation in Presence of Gravity

Simultaneous capillary dominated displacement of the wetting and non-wetting phases are processes of interest in many disciplines including modeling of the penetration of polluting liquids in hydrology or the secondary migration in petroleum reservoir engineering. Percolation models and in particular invasion percolation is well suited to characterize the slow immiscible displaceme...

The scaling limit of the incipient infinite cluster in high-dimensional percolation. I. Critical exponents

This is the first of two papers on the critical behaviour of bond percolation models in high dimensions. In this paper, we obtain strong joint control of the critical exponents η and δ, for the nearest-neighbour model in very high dimensions d 6 and for sufficiently spreadout models in all dimensions d > 6. The exponent η describes the low frequency behaviour of the Fourier transform of the cri...

Scaling limit of simple random walk on supercritical percolation clusters