Related papers: MSO 0-1 law for recursive random trees

We introduce the notion of a distributive law between a relative monad and a monad. We call this a relative distributive law and define it in any 2-category $\mathcal{K}$. In order to do that, we introduce the 2-category of relative monads…

We analyze the tree-level 2->2 scattering of massive spin-2 bosons in a theory with only relevant and marginal operators and extract the sum rules on the coupling constants and masses required to achieve tree-level unitarity to very high…

The monadic second-order theory of trees allows quantification over elements and over arbitrary subsets. We classify the class of trees with respect to the question: does a tree T have definable Skolem functions (by a monadic formula with…

We compare the expressiveness of two extensions of monadic second-order logic (MSO) over the class of finite structures. The first, counting monadic second-order logic (CMSO), extends MSO with first-order modulo-counting quantifiers,…

A tree is pathwise-random if all of its paths are Martin-Lof random. We show that (a) no weakly 2-random real computes a perfect pathwise-random tree; it follows that the class of perfect pathwise-random trees is null, with respect to any…

We investigate algorithms to find the first vertex in large trees generated by either the uniform attachment or preferential attachment model. We require the algorithm to output a set of $K$ vertices, such that, with probability at least…

We consider the problem of computing the measure of a regular language of infinite binary trees. While the general case remains unsolved, we show that the measure of a language defined by a first-order formula with no descendant relation or…

We prove a law of large numbers in terms of complete convergence of independent random variables taking values in increments of monotone functions, with convergence uniform both in the initial and the final time. The result holds also for…

One class of random walks with infinite memory, so called elephant random walks, are simple models describing anomalous diffusion. We present a surprising connection between these models and bond percolation on random recursive trees. We…

Complex networks in different areas exhibit degree distributions with heavy upper tail. A preferential attachment mechanism in a growth process produces a graph with this feature. We herein investigate a variant of the simple preferential…

We show that an algorithmic construction of sequences of recursive trees leads to a direct proof of the convergence of random recursive trees in an associated Doob-Martin compactification; it also gives a representation of the limit in…

We say that a convergence law holds for a sequence of random combinatorial objects if, for any first-order sentence $\varphi$, the proportion of objects satisfying $\varphi$ converges to a limiting value as the size of the objects tends to…

We are interested in the asymptotics of random trees built by linear preferential attachment, also known in the literature as Barab\'asi-Albert trees or plane-oriented recursive trees. We first prove a conjecture of Bubeck, Mossel \& R\'acz…

This manuscript establishes several sufficient conditions for the validity of both the reverse order law and forward order law for NDMPI. Additionally, some characterization of the reverse order law of the NDMPI is obtained. We also explore…

We study the parking process on the random recursive tree. We first prove that although the random recursive tree has a non-degenerate Benjamini--Schramm limit, the phase transition for the parking process appears at density $0$. We then…

We investigate scaling limits of trees built by uniform attachment with freezing, which is a variant of the classical model of random recursive trees introduced in a companion paper. Here vertices are allowed to freeze, and arriving…

We consider injective first-order interpretations that input and output trees of bounded height. The corresponding functions have polynomial output size, since a first-order interpretation can use a k-tuple of input nodes to represent a…

The first-order theory of a string automatic structure is known to be decidable, but there are examples of string automatic structures with nonelementary first-order theories. We prove that the first-order theory of a string automatic…

A multi-type preferential attachment tree is introduced, and studied using general multi-type branching processes. For the $p$-type case we derive a framework for studying the tree where a type $i$ vertex generates new type $j$ vertices…

The first main result of this paper is that the law of the (rescaled) two-dimensional uniform spanning tree is tight in a space whose elements are measured, rooted real trees continuously embedded into Euclidean space. Various properties of…