Related papers: Asymptotic normality of consecutive patterns in pe…
We discuss a new notion of pattern avoidance motivated by the operad theory: pattern avoidance in planar labelled trees. It is a generalisation of various types of consecutive pattern avoidance studied before: consecutive patterns in words,…
We study branching random walks on Cayley graphs. A first result is that the trace of a transient branching random walk on a Cayley graph is a.s. transient for the simple random walk. In addition, it has a.s. critical percolation…
We study (asymmetric) $U$-statistics based on a stationary sequence of $m$-dependent variables; moreover, we consider constrained $U$-statistics, where the defining multiple sum only includes terms satisfying some restrictions on the gaps…
We investigate the statistics of three kinds of records associated with planar random walks, namely diagonal, simultaneous and radial records. The mean numbers of these records grow as universal power laws of time, with respective exponents…
We study a universal object for the genealogy of a sample in populations with mutations: the critical birth-death process with Poissonian mutations, conditioned on its population size at a fixed time horizon. We show how this process arises…
A Bernoulli random walk is a random trajectory starting from 0 and having i.i.d. increments, each of them being $+1$ or -1, equally likely. The other families cited in the title are Bernoulli random walks under various conditionings. A peak…
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…
Our work studies the enumeration and random generation of unlabeled combinatorial classes of unrooted graphs. While the technique of vertex pointing provides a straightforward procedure for analyzing a labeled class of unrooted graphs by…
Let x and y be chosen uniformly in a graph G. We find the limiting distribution of the length of a loop-erased random walk from x to y on a large class of graphs that include the discrete torus in dimensions 5 and above. Moreover, on this…
We consider a random permutation drawn from the set of 132-avoiding permutations of length $n$ and show that the number of occurrences of another pattern $\sigma$ has a limit distribution, after scaling by $n^{\lambda(\sigma)/2}$ where…
We study the evolution of a random walker on a conservative dynamic random environment composed of independent particles performing simple symmetric random walks, generalizing results of [16] to higher dimensions and more general transition…
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…
Using the recently developed notion of permutation limits this paper derives the limiting distribution of the number of fixed points and cycle structure for any convergent sequence of random permutations, under mild regularity conditions.…
We consider growing random recursive trees in random environment, in which at each step a new vertex is attached (by an edge of a random length) to an existing tree vertex according to a probability distribution that assigns the tree…
In [1], the authors consider a random walk $(Z_{n,1},\ldots,Z_{n,K+1})\in \mathbb{Z}^{K+1}$ with the constraint that each coordinate of the walk is at distance one from the following one. A functional central limit theorem for the first…
We investigate the directed random walk on hierarchic trees. Two cases are investigated: random variables on deterministic trees with a continuous branching, and random variables on the trees constructed trough the random branching process.…
Search trees are fundamental data structures in computer science. We study functionals on random search trees that satisfy recurrence relations of a simple additive form. Many important functionals including the space requirement, internal…
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…
We study spanning trees on Sierpinski graphs (i.e., finite approximations to the Sierpinski gasket) that are chosen uniformly at random. We construct a joint probability space for uniform spanning trees on every finite Sierpinski graph and…
In the last decade a huge amount of articles has been published studying pattern avoidance on permutations. From the point of view of enumeration, typically one tries to count permutations avoiding certain patterns according to their…