Related papers: Thermodynamic Limit for Large Random Trees

This work will appear as a chapter in a forthcoming volume titled "Topics in Probabilistic Graph Theory". A theory of scaling limits for random graphs has been developed in recent years. This theory gives access to the large-scale geometric…

Let $\tau$n be a random tree distributed as a Galton-Watson tree with geometric offspring distribution conditioned on {Zn = an} where Zn is the size of the n-th generation and (an, n $\in$ N *) is a deterministic positive sequence. We study…

Consider a family of random ordered graph trees $(T_n)_{n\geq 1}$, where $T_n$ has $n$ vertices. It has previously been established that if the associated search-depth processes converge to the normalised Brownian excursion when rescaled…

We prove that for any fixed $k$, the probability that a random vertex of a random increasing plane tree is of rank $k$, that is, the probability that a random vertex is at distance $k$ from the leaves, converges to a constant $c_k$ as the…

Suppose that $G_j$ is a sequence of finite connected planar graphs, and in each $G_j$ a special vertex, called the root, is chosen randomly-uniformly. We introduce the notion of a distributional limit $G$ of such graphs. Assume that the…

We provide the law of large numbers for roots of finite free multiplicative convolution of polynomials which have only non-negative real roots. Moreover, we study the empirical root distributions of limit polynomials obtained through the…

We introduce the notion of a restricted exchangeable partition of $\mathbb{N}$. We obtain integral representations, consider associated fragmentations, embeddings into continuum random trees and convergence to such limit trees. In…

We prove that, in the random stirring model of parameter T on an infinite rooted tree each of whose vertices has at least two offspring, infinite cycles exist almost surely, provided that T is sufficiently high. In the appendices, the bound…

Local convergence of bounded degree graphs was introduced by Benjamini and Schramm. This result was extended further by Lyons to bounded average degree graphs. In this paper we study the convergence of random tree sequences with given…

A symbolic-computational algorithm, fully implemented in Maple, is described, that computes explicit expressions for generating functions that enable the efficient computations of the expectation, variance, and higher moments, of the random…

We consider ferromagnetic Ising models on graphs that converge locally to trees. Examples include random regular graphs with bounded degree and uniformly random graphs with bounded average degree. We prove that the "cavity" prediction for…

We consider random binary trees that appear as the output of certain standard algorithms for sorting and searching if the input is random. We introduce the subtree size metric on search trees and show that the resulting metric spaces…

We consider a marking procedure of the vertices of a tree where each vertex is marked independently from the others with a probability that depends only on its out-degree. We prove that a critical Galton-Watson tree conditioned on having a…

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…

We prove for Gibbs-Markov maps that the number of visits to a sequence of shrinking sets with bounded cylindrical lengths converges in distribution to a Poisson law. Applying to continued fractions, this result extends Doeblin's Poisson…

The hierarchical and recursive expressive capability of rooted trees is applicable to represent statistical models in various areas, such as data compression, image processing, and machine learning. On the other hand, such hierarchical…

We examine the minimization of information entropy for measures on the phase space of bounded domains, subject to constraints that are averages of grand canonical distributions. We describe the set of all such constraints and show that it…

We introduce the notion of a hereditary property for rooted real trees and we also consider reduction of trees by a given hereditary property. Leaf-length erasure, also called trimming, is included as a special case of hereditary reduction.…

We investigate the statistics of trees grown from some initial tree by attaching links to preexisting vertices, with attachment probabilities depending only on the valence of these vertices. We consider the asymptotic mass distribution that…

We prove an invariance principle for a general class of continuous time critical branching processes with finite variance (non-local) branching mechanism. We show that the genealogical trees, viewed as random compact metric measure spaces,…