Related papers: Schr\"oder's problems and scaling limits of random…

We prove limit theorems for sums of functions of subtrees of binary search trees and random recursive trees. In particular, we give simple new proofs of the fact that the number of fringe trees of size $ k=k_n $ in the binary search tree…

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 give an explicit construction of the scaling limit of the minimum spanning tree of the complete graph. The limit object is described using a recursive construction involving the convex minorants of a Brownian motion with parabolic drift…

We show that the expected size of the maximum agreement subtree of two $n$-leaf trees, uniformly random among all trees with the shape, is $\Theta(\sqrt{n})$. To derive the lower bound, we prove a global structural result on a decomposition…

These are the notes for a series of lectures given at the Clay Mathematical Institute Summer School in Buzios, July 11 - August 7, 2010. We review some of the recent aspects of scaling limits of random trees and planar maps, in particular…

We study the joint distribution of the number of occurrences of members of a collection of nonoverlapping motifs in digital data. We deal with finite and countably infinite collections. For infinite collections, the setting requires that we…

This article is motivated by the following satisfiability question: pick uniformly at random an and/or Boolean expression of length n, built on a set of k_n Boolean variables. What is the probability that this expression is satisfiable?…

We study the space requirements of a sorting algorithm where only items that at the end will be adjacent are kept together. This is equivalent to the following combinatorial problem: Consider a string of fixed length n that starts as a…

We consider large uniform labeled random graphs in different classes with prescribed decorations in their modular decomposition. Our main result is the estimation of the number of copies of every graph as an induced subgraph. As a…

We prove limit laws for the number of occurrences of a pattern on the fringe of a ranked tree-child network which is picked uniformly at random. Our results extend the limit law for cherries proved by Bienvenu et al. (2022). For patterns of…

We determine upper and lower bounds for the number of maximum matchings (i.e., matchings of maximum cardinality) $m(T)$ of a tree $T$ of given order. While the trees that attain the lower bound are easily characterised, the trees with…

We consider the cubic nonlinear fourth-order Schr\"odinger equation \[ i\partial_t u - \Delta^2 u + \mu \Delta u = \pm |u|^2 u, \quad \mu \geq 0 \] on $\mathbb{R}^N, N \geq 5$ with random initial data. We prove almost sure local…

We obtain the Brownian net of Sun and Swart (2008) as the scaling limit of the paths traced out by a system of continuous (one-dimensional) space and time branching and coalescing random walks. This demonstrates a certain universality of…

We study a model of random $\mathcal{R}$-enriched trees that is based on weights on the $\mathcal{R}$-structures and allows for a unified treatment of a large family of random discrete structures. We establish distributional limits…

In this paper, we compute the distribution of the first letter statistic on nine avoidance classes of permutations corresponding to two pairs of patterns of length four. In particular, we show that the distribution is the same for each…

We consider the tributary structure of Howard's drainage model studied by Gangopadhyay et. al. Conditional on the event that the tributary survives up to time $n$, we show that, as a sequence of random metric spaces, scaled tributary…

We consider the enumeration of plane trees (rooted ordered trees) whose vertices are colored according to a specific coloring rule that prescribes which possible pairs of colors can occur as the colors of a parent vertex and its child. This…

We establish large deviations for dynamical Schr\"{o}dinger problems driven by perturbed Brownian motions when the noise parameter tends to zero. Our results show that Schr\"{o}dinger bridges charge exponentially small masses outside the…

For large $n$, take a random $n \times n$ permutation matrix and its associated discrete copula $X_n$. For $a, b = 0, 1, \ldots, n$, let $y_n(\frac{a}{n},\frac{b}{n}) = \frac{1}{n} ( X_{a,b} - \frac{ab}{n} )$; define $y_n: [0,1]^2 \to R$ by…

There are several good reasons you might want to read about uniform spanning trees, one being that spanning trees are useful combinatorial objects. Not only are they fundamental in algebraic graph theory and combinatorial geometry, but they…