Related papers: Growth dichotomy for unimodular random rooted tree…

Let $d\ge 3$ be a fixed integer. Let $y:= y(p)$ be the probability that the root of an infinite $d$-regular tree belongs to an infinite cluster after $p$-bond-percolation. We show that for every constants $b,\alpha>0$ and $1<\lambda< d-1$,…

We provide sufficient conditions for a regular graph $G$ of growing degree $d$, guaranteeing a phase transition in its random subgraph $G_p$ similar to that of $G(n,p)$ when $p\cdot d\approx 1$. These conditions capture several well-studied…

For every sufficiently well-behaved function $g:\mathbb{R}_{\ge 0}\rightarrow\mathbb{R}_{\ge 0}$ that grows at least linearly and at most exponentially we construct a tree $T$ of uniform volume growth $g$, that is, $$C_1\cdot g(r/4)\le…

We study non-uniform percolation in a two-dimensional cluster growth model with multiple seeds. With increasing concentration of seeds, the percolation threshold is found to increase monotonically, while the exponents for correlation…

For a unimodular random graph $(G,\rho)$, we consider deformations of its intrinsic path metric by a (random) weighting of its vertices. This leads to the notion of the conformal growth exponent of $(G,\rho)$, which is the best asymptotic…

We construct a unimodular random rooted graph with maximal degree $d\geq 3$ and upper growth rate $d-1$, which does not have a growth rate. Ab\'ert, Fraczyk and Hayes showed that for a unimodular random tree, if the upper growth rate is at…

The critical phase of bond percolation on the random growing tree is examined. It is shown that the root cluster grows with the system size $N$ as $N^\psi$ and the mean number of clusters with size $s$ per node follows a power function $n_s…

In invasion percolation, the edges of successively maximal weight (the outlets) divide the invasion cluster into a chain of ponds separated by outlets. On the regular tree, the ponds are shown to grow exponentially, with law of large…

Consider percolation on $T\times \mathbb{Z}^d$, the product of a regular tree of degree $k\geq 3$ with the hypercubic lattice $\mathbb{Z}^d$. It is known that this graph has $0<p_c<p_u<1$, so that there are non-trivial regimes in which…

Consider an infinite planar graph with uniform polynomial growth of degree d > 2. Many examples of such graphs exhibit similar geometric and spectral properties, and it has been conjectured that this is necessary. We present a family of…

For each integer $k \geq 2$, we introduce a sequence of $k$-ary discrete trees constructed recursively by choosing at each step an edge uniformly among the present edges and grafting on "its middle" $k-1$ new edges. When $k=2$, this…

We prove that a uniform, rooted unordered binary tree with $n$ vertices has the Brownian continuum random tree as its scaling limit for the Gromov-Hausdorff topology. The limit is thus, up to a constant factor, the same as that of uniform…

We analyze a simple model for growing tree networks and find that although it never percolates, there is an anomalously large cluster at finite size. We study the growth of both the maximal cluster and the cluster containing the original…

In many interacting particle systems, relaxation to equilibrium is thought to occur via the growth of 'droplets', and it is a question of fundamental importance to determine the critical length at which such droplets appear. In this paper…

We destroy a finite tree of size $n$ by cutting its edges one after the other and in uniform random order. Informally, the associated cut-tree describes the genealogy of the connected components created by this destruction process. We…

We continue the study of the level-set percolation of the discrete Gaussian free field (GFF) on regular trees in the critical regime, initiated in arXiv:2302.02753. First, we derive a sharp asymptotic estimate for the probability that the…

This paper extends the study of fringe trees in random plane trees with a given degree statistic. While previous work established the asymptotic normality of the count of fringe trees isomorphic to a fixed tree, we investigate the case…

This paper introduces a new combinatorial framework for modeling the growth of binary trees through a discrete evolution process that incorporates a growing rule and an extinction rule. Building upon the theory of increasingly labeled…

In this paper, we establish a coupling lemma for standard families in the setting of piecewise expanding interval maps with countably many branches. Our method merely requires that the expanding map satisfies Chernov's one-step expansion at…

We define some new sequences of recursively constructed random combinatorial trees, and show that, after properly rescaling graph distance and equipping the trees with the uniform measure on vertices, each sequence converges almost surely…