Related papers: Local limit of the fixed point forest

The Robinson-Foulds (RF) distance is by far the most widely used measure of dissimilarity between trees. Although the distribution of these distances has been investigated for twenty years, an algorithm that is explicitly polynomial time…

We construct a stationary random tree, embedded in the upper half plane, with prescribed offspring distribution and whose vertices are the atoms of a unit Poisson point process. This process which we call Hammersley's tree process extends…

In a uniform random permutation \Pi of [n] := {1,2,...,n}, the set of elements k in [n-1] such that \Pi(k+1) = \Pi(k) + 1 has the same distribution as the set of fixed points of \Pi that lie in [n-1]. We give three different proofs of this…

Random events in space and time often exhibit a locally dependent structure. When the events are very rare and dependent structure is not too complicated, various studies in the literature have shown that Poisson and compound Poisson…

We consider statistics on permutations chosen uniformly at random from fixed parabolic double cosets of the symmetric group. We show that the distribution of fixed points is asymptotically Poisson and establish central limit theorems for…

This paper initiates a limit theory of permutation valued processes, building on the recent theory of permutons. We apply this to study the asymptotic behaviour of random sorting networks. We prove that the Archimedean path, the conjectured…

We prove limit theorems for the number of fixed points occurring in a random pattern-avoiding permutation distributed according to a one-parameter family of biased distributions. The bias parameter exponentially tilts the distribution…

Motivated by online recommendation systems, we study a family of random forests. The vertices of the forest are labeled by integers. Each non-positive integer $i\le 0$ is the root of a tree. Vertices labeled by positive integers $n \ge 1$…

Kesten and Lee [36] proved that the total length of a minimal spanning tree on certain random point configurations in $\mathbb{R}^d$ satisfies a central limit theorem. They also raised the question: how to make these results quantitative?…

We discuss a notion of convergence for binary trees that is based on subtree sizes. In analogy to recent developments in the theory of graphs, posets and permutations we investigate some general aspects of the topology, such as a…

We consider finding a counterfactual explanation for a classification or regression forest, such as a random forest. This requires solving an optimization problem to find the closest input instance to a given instance for which the forest…

We study the asymptotic behavior of short cycles of random permutations with cycle weights. More specifically, on a specially constructed metric space whose elements encode all possible cycles, we consider a point process containing all…

We study the random geometry of first passage percolation on the complete graph equipped with independent and identically distributed edge weights, continuing the program initiated by Bhamidi and van der Hofstad [9]. We describe our results…

Consider a graph on randomly scattered points in an arbitrary space, with two points $x,y$ connected with probability $\phi(x,y)$. Suppose the number of points is large but the mean number of isolated points is $O(1)$. We give general…

We study fragmentation of a random recursive tree into a forest by repeated removal of nodes. The initial tree consists of N nodes and it is generated by sequential addition of nodes with each new node attaching to a randomly-selected…

We define some new sequences of recursively constructed random combinatorial trees, and show that, after properly rescaling graph distance and equipping the trees with the uniform measure on vertices, each sequence converges almost surely…

In this article, we consider a Branching Random Walk on the real line. The genealogical structure is assumed to be given through a supercritical branching process in the i.i.d. environment and satisfies the Kesten-Stigum condition. The…

Consider a uniformly chosen random reduced decomposition of the longest element in the symmetric group. It is known that the location of the first transposition in this decomposition converges to the semicircle distribution. In this note we…

Let $\eta_t$ be a Poisson point process of intensity $t\geq 1$ on some state space $\Y$ and $f$ be a non-negative symmetric function on $\Y^k$ for some $k\geq 1$. Applying $f$ to all $k$-tuples of distinct points of $\eta_t$ generates a…

We establish a novel bijective encoding that represents permutations as forests of decorated (or enriched) trees. This allows us to prove local convergence of uniform random permutations from substitution-closed classes satisfying a…