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We study a symmetric random walk (RW) in one spatial dimension in environment, formed by several zones of finite width, where the probability of transition between two neighboring points and corresponding diffusion coefficient are…
In the critical beta-splitting model of a random $n$-leaf rooted tree, clades are recursively split into sub-clades, and a clade of $m$ leaves is split into sub-clades containing $i$ and $m-i$ leaves with probabilities $\propto 1/(i(m-i))$.…
Permutations that avoid given patterns are among the most classical objects in combinatorics and have strong connections to many fields of mathematics, computer science and biology. In this paper we study the scaling limits of a random…
Coalescing simple random walks in the plane form an infinite tree. A natural directed distance on this tree is given by the number of jumps between branches when one is only allowed to move in one direction. The Brownian web distance is the…
We consider a Tamari interval of size $n$ (i.e., a pair of Dyck paths which are comparable for the Tamari relation) chosen uniformly at random. We show that the height of a uniformly chosen vertex on the upper or lower path scales as…
We study the statistics of height and balanced height in the binary search tree problem in computer science. The search tree problem is first mapped to a fragmentation problem which is then further mapped to a modified directed polymer…
We obtain assumption-free, non-asymptotic, uniform bounds on the product of the height and the width of uniformly random trees with a given degree sequence, conditioned Bienaym\'e trees and simply generated trees. We show that for a tree of…
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 provide a new construction of Brownian disks in terms of forests of continuous random trees equipped with nonnegative labels corresponding to distances from a distinguished point uniformly distributed on the boundary of the disk. This…
We describe the asymptotic behaviour of large degrees in random hyperbolic graphs, for all values of the curvature parameter $ \alpha$. We prove that, with high probability, the node degrees satisfy the following ordering property: the…
We address the question whether the sequence of areas between coalescing random walkers displays multiscaling and in the process calculate the second moment as well as the two point correlation function exactly. The scaling of higher order…
We consider random geometric graphs on the plane characterized by a non-uniform density of vertices. In particular, we introduce a graph model where $n$ vertices are independently distributed in the unit disc with positions, in polar…
We consider uniform random cographs (either labeled or unlabeled) of large size. Our first main result is the convergence towards a Brownian limiting object in the space of graphons. We then show that the degree of a uniform random vertex…
We study periodic Brownian paths, wrapped around the surface of a cylinder. One characteristic of such a path is its width square, $w^2$, defined as its variance. Though the average of $w^2$ over all possible paths is well known, its full…
Asymptotic estimates of the hitting distribution of a long segment on the real axis for two dimensional random walks on ${\bf Z}^2$ of zero mean and finite variances are obtained: some are general and exhibit its apparent similarity to the…
We consider the model of random planar maps of size $n$ biased by a weight $u>0$ per $2$-connected block, and the closely related model of random planar quadrangulations of size $n$ biased by a weight $u>0$ per simple component. We exhibit…
We study fluctuations of an ensemble of $N$ independent particles undergoing anomalous diffusion with random renewal resetting. The anomalous diffusion is modeled by the scaled Brownian motion (sBm): a Gaussian process, characterized by a…
We consider a class of real numbers, a subset of irrational numbers and certain mathematical constants, for which the elements in the simple continued fraction appears to be random. As an illustrative example, one can consider $\pi = \{x_0,…
If L is a partition of n, the rank of L is the size of the largest part minus the number of parts. Under the uniform distribution on partitions, Bringmann, Mahlburg, and Rhoades showed that the rank statistic has a limiting distribution. We…
In this paper we introduce a new model of random spanning trees that we call choice spanning trees, constructed from so-called choice random walks. These are random walks for which each step is chosen from a subset of random options,…