Related papers: Recurrence on the average on trees
We develop a new approach to recurrence and the existence of non-constant harmonic functions on infinite weighted graphs. The approach is based on the capacity of subsets of metric boundaries with respect to intrinsic metrics. The main tool…
Measuring the complexity of tree structures can be beneficial in areas that use tree data structures for storage, communication, and processing purposes. This complexity can then be used to compress tree data structures to their…
We study the evolution of branching trees embedded in Euclidean spaces with suppressed branching of spatially close nodes. This cooperative branching process accounts for the effect of overcrowding of nodes in the embedding space and mimics…
We study Bernoulli bond percolation on a random recursive tree of size $n$ with percolation parameter $p(n)$ converging to $1$ as $n$ tends to infinity. The sizes of the percolation clusters are naturally stored in a tree. We prove…
Exploiting a bijective correspondence between planar quadrangulations and well-labeled trees, we define an ensemble of infinite surfaces as a limit of uniformly distributed ensembles of quadrangulations of fixed finite volume. The limit…
A quasiconformal tree is a doubling (compact) metric tree in which the diameter of each arc is comparable to the distance of its endpoints. We show that for each integer $n\geq 2$, the class of all quasiconformal trees with uniform branch…
The properties of scale-free random trees are investigated using both preconditioning on non-extinction and fixed size averages, in order to study the thermodynamic limit. The scaling form of volume probability is found, the connectivity…
In this paper we investigate the geometric properties of quasi-trees, and prove some equivalent criteria. We give a general construction of a tree that approximates the ends of a geodesic space, and use this to prove that every quasi-tree…
Random-cluster measures on infinite regular trees are studied in conjunction with a general type of `boundary condition', namely an equivalence relation on the set of infinite paths of the tree. The uniqueness and non-uniqueness of…
We identify a structural pattern in the construction of known infinite families of trees whose independence polynomials are not log-concave. Using this pattern and properties of polynomial ring ideals, we derive linear recurrences for these…
We study the recurrence and ergodicity for the billiard on noncompact polygonal surfaces with a free, cocompact action of $\Z$ or $\Z^2$. In the $\Z$-periodic case, we establish criteria for recurrence. In the more difficult $\Z^2$-periodic…
Consider a random real tree whose leaf set, or boundary, is endowed with a finite mass measure. Each element of the tree is further given a type, or allele, inherited from the most recent atom of a random point measure…
For a tree Markov random field non-reconstruction is said to hold if as the depth of the tree goes to infinity the information that a typical configuration at the leaves gives about the value at the root goes to zero. The distribution of…
We show that for lazy simple random walks on finite spherically symmetric trees, the ratio of the mixing time and the relaxation time is bounded by a universal constant. Consequently, lazy simple random walks on any sequence of finite…
We give a simple proof that the straightforward generalisation of clique-width to arbitrary structures can be unbounded on structures of bounded tree-width. This can be corrected by allowing fusion of elements.
We consider Gibbs distributions on finite random plane trees with bounded branching. We show that as the order of the tree grows to infinity, the distribution of any finite neighborhood of the root of the tree converges to a limit. We…
We analyze the finite sample mean squared error (MSE) performance of regression trees and forests in the high dimensional regime with binary features, under a sparsity constraint. We prove that if only $r$ of the $d$ features are relevant…
Minimal spanning trees on infinite vertex sets are investigated. A criterion for minimality of a spanning tree having a finite length is obtained, which generalizes the corresponding classical result for finite sets. It is given an analytic…
A uniform recursive tree on $n$ vertices is a random tree where each possible $(n-1)!$ labeled recursive rooted tree is selected with equal probability. In this paper we introduce and study weighted trees, a non-uniform recursive tree model…
We give new general formulas for the asymptotics of the number of spanning trees of a large graph. A special case answers a question of McKay (1983) for regular graphs. The general answer involves a quantity for infinite graphs that we call…