Related papers: Fires on large recursive trees
An infection spreads in a binary tree of height n as follows: initially, each leaf is either infected by one of k states or it is not infected at all. The infection state of each leaf is independently distributed according to a probability…
We examine an interacting particle system on trees commonly referred to as the frog model. For its initial state, it begins with a single active particle at the root and i.i.d. $\mathrm{Poiss}(\lambda)$ many inactive particles at each…
Chip-firing is a combinatorial game played on an undirected graph in which we place chips on vertices. We study chip-firing on an infinite binary tree in which we add a self-loop to the root to ensure each vertex has degree 3. A vertex can…
We prove a law of large numbers for the range of rotor walks with random initial configuration on regular trees and on Galton-Watson trees. More precisely, we show that on the classes of trees under consideration, even in the case when the…
We study the scaling limit of random forest with prescribed degree sequence in the regime that the largest tree consists of all but a vanishing fraction of nodes. We give a description of the limit of the forest consisting of the small…
The purpose of this paper is to analyze certain statistics of a recently introduced non-uniform random tree model, biased recursive trees. This model is based on constructing a random tree by establishing a correspondence with non-uniform…
Let $F(N,m)$ denote a random forest on a set of $N$ vertices, chosen uniformly from all forests with $m$ edges. Let $F(N,p)$ denote the forest obtained by conditioning the Erdos-Renyi graph $G(N,p)$ to be acyclic. We describe scaling limits…
The recursive and hierarchical structure of full rooted trees is applicable to represent statistical models in various areas, such as data compression, image processing, and machine learning. In most of these cases, the full rooted tree is…
The firefighter problem is a monotone dynamic process in graphs that can be viewed as modeling the use of a limited supply of vaccinations to stop the spread of an epidemic. In more detail, a fire spreads through a graph, from burning…
Motivated by questions in social networks, distributed computing and probabilistic combinatorics, the last few years have seen increasing interest in network evolution models where new vertices entering the system need to make decisions…
We show, under natural conditions, that uniform rooted trees with fixed degree sequence converge after renormalization toward inhomogeneous continuum random trees (ICRT). We also provide a sharp upper-bound for the tail of their heights. We…
In this work, we study the color discrepancy of spanning trees in random graphs. We show that for the Erd\H{o}s-R\'enyi random graph $G(n,p)$ with $p$ above the connectivity threshold, the following holds with high probability: in every…
We study effective randomness-preserving transformations of path-incompressible trees. Some path-incompressible trees with infinitely many paths do not compute perfect path-random trees with computable oracle-use. Sparse perfect…
The contact process is a simple model for the spread of an infection in a structured population. We investigate the case when the underlying structure evolves dynamically as a degree-dependent dynamical percolation model. Starting with a…
In this paper, we study a regular rooted coloured tree with random labels assigned to its edges, where the distribution of the label assigned to an edge depends on the colours of its endpoints. We obtain some new results relevant to this…
We use an infinite $k$-ary tree with a self-loop at the root as our underlying graph. We consider a chip-firing process starting with $N$ chips at the root. We describe the stable configurations. We calculate the number of fires for each…
In this paper we show that all infinite trees which have bounded coordination and whose surface is negligible with respect to the volume in the limit of large distances (so that they can be embedded in a finite-dimensional euclidean space)…
Limiting distributions are derived for the sparse connected components that are present when a random graph on $n$ vertices has approximately $\half n$ edges. In particular, we show that such a graph consists entirely of trees, unicyclic…
We study a random recursive tree model featuring complete redirection called the random friend tree and introduced by Saram\"aki and Kaski. Vertices are attached in a sequential manner one by one by selecting an existing target vertex and…
We prove the well-posedness of a differential equation that describes the evolution of the large-system limit of the empirical age measure in the mean field forest fire model of R\'ath and T\'oth (arXiv:0808.2116). This forest fire model is…