Critical Percolation and the Incipient Infinite Cluster on Galton-Watson Trees

In this paper, we study the random walk on a supercritical branching process with an uncountable and unbounded set of types supported on the $d$-regular tree $\mathbb{T}_d$ ($d\geq 3$), namely the cluster $\mathcal{C}_\circ^h$ of the root in the level set of the Gaussian Free Field (GFF) above an arbitrary value $h\in (-\infty, h_{\star})$. The value $h_{\star}\in (0,\infty)$ is the percolation threshold; in particular, $\mathcal{C}_\circ^h$ is infinite with positive probabil...

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 $r\geq2$ 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 analyze simple random walk on a supercritical Galton-Watson tree, where the walk is conditioned to return to the root at time $2n$. Specifically, we establish the asymptotic order (up to a constant factor) as $n\to\infty$, of the maximal displacement from the root. Our results, which are shown to hold for almost surely every surviving tree $T$ (subject to some mild moment conditions), are broken up into two cases. When the offspring distribution takes a value less than or ...

Let ${\cal G}$ be the incipient infinite cluster (IIC) for percolation on a homogeneous tree of degree $n_0+1$. We obtain estimates for the transition density of the continuous time simple random walk $Y$ on ${\cal G}$; the process satisfies anomalous diffusion and has spectral dimension 4/3.

We observe the Galton-Watson Branching Processes. Limit properties of transition functions and their convergence to invariant measures are investigated.

We consider Bienaym\'e-Galton-Watson trees in random environment, where each generation $k$ is attributed a random offspring distribution $\mu_k$, and $(\mu_k)_{k\geq 0}$ is a sequence of independent and identically distributed random probability measures. We work in the ``strictly critical'' regime where, for all $k$, the average of $\mu_k$ is assumed to be equal to $1$ almost surely, and the variance of $\mu_k$ has finite expectation. We prove that, for almost all realizati...

We study a particular type of subcritical Galton--Watson trees, which are called non-generic trees in the physics community. In contrast with the critical or supercritical case, it is known that condensation appears in certain large conditioned non-generic trees, meaning that with high probability there exists a unique vertex with macroscopic degree comparable to the total size of the tree. Using recent results concerning subexponential distributions, we investigate this phen...

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 eta and delta, for the nearest-neighbour model in very high dimensions d>>6 and for sufficiently spread-out models in all dimensions d>6. The exponent eta describes the low frequency behaviour of the Fourier transform of the critical two-point connectivity function, while delta describes the behaviour ...

We present a new pruning procedure on discrete trees by adding marks on the nodes of trees. This procedure allows us to construct and study a tree-valued Markov process $\{{\cal G}(u)\}$ by pruning Galton-Watson trees and an analogous process $\{{\cal G}^*(u)\}$ by pruning a critical or subcritical Galton-Watson tree conditioned to be infinite. Under a mild condition on offspring distributions, we show that the process $\{{\cal G}(u)\}$ run until its ascension time has a repr...

We give a unified treatment of the limit, as the size tends to infinity, of simply generated random trees, including both the well-known result in the standard case of critical Galton--Watson trees and similar but less well-known results in the other cases (i.e., when no equivalent critical Galton--Watson tree exists). There is a well-defined limit in the form of an infinite random tree in all cases; for critical Galton--Watson trees this tree is locally finite but for the ot...