Related papers: Almost sure asymptotics for the random binary sear…

We conduct the multifractal analysis of the level sets of the asymptotic behavior of almost additive continuous potentials $(\phi_n)_{n=1}^\infty$ on a topologically mixing subshift of finite type $X$ endowed itself with a metric associated…

We consider maximum rooted tree extension counts in random graphs, i.e., we consider M_n = \max_v X_v where X_v counts the number of copies of a given tree in G_{n,p} rooted at vertex v. We determine the asymptotics of M_n when the random…

We prove almost sure strong asymptotic freeness of i.i.d. random unitaries with the following law: sample a Haar unitary matrix of dimension $n$ and then send this unitary into an irreducible representation of $U(n)$. The strong convergence…

Inspired by Stufler's recent probabilistic proof of Otter's asymptotic number of unlabeled trees, we revisit work of Palmer and Schwenk, and study unlabeled forests from a probabilistic point of view. We show that the number of trees in a…

The Colijn--Plazzotta ranking is a bijective encoding of the unlabeled binary rooted trees with positive integers. We show that the rank $f(t)$ of a tree $t$ is closely related to its height $h$, the length of the longest path from a leaf…

We prove that a random automaton with $n$ states and any fixed non-singleton alphabet is synchronizing with high probability (modulo an unpublished result about unique highest trees of random graphs). Moreover, we also prove that the…

Motivated by real-world machine learning applications, we analyze approximations to the non-asymptotic fundamental limits of statistical classification. In the binary version of this problem, given two training sequences generated according…

Suppose the edges of the complete graph on $n$ vertices are assigned a uniformly chosen random ordering. Let $X$ denote the corresponding number of Hamiltonian paths that are increasing in this ordering. It was shown in a recent paper by…

We prove that if a tree $T$ has $n$ vertices and maximum degree at most $\Delta$, then a copy of $T$ can almost surely be found in the random graph $\mathcal{G}(n,\Delta\log^5 n/n)$.

We discuss asymptotics for large random planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable distribution with index $\alpha\in(1,2)$. When the number $n$ of…

We establish a new asymptotic formula for the number of polynomials of degree $n$ with $k$ prime factors over a finite field $\mathbb{F}_q$. The error term tends to $0$ uniformly in $n$ and in $q$, and $k$ can grow beyond $\log n$.…

Let $f$ be a Rademacher or a Steinhaus random multiplicative function. Let $\varepsilon>0$ small. We prove that, as $x\rightarrow +\infty$, we almost surely have $$\bigg|\sum_{\substack{n\leq x\\…

This extended abstract is dedicated to the analysis of the height of non-plane unlabelled rooted binary trees. The height of such a tree chosen uniformly among those of size $n$ is proved to have a limiting theta distribution, both in a…

We consider a branching random walk for which the maximum position of a particle in the n'th generation, M_n, has zero speed on the linear scale: M_n/n --> 0 as n --> infinity. We further remove ("kill") any particle whose displacement is…

We present a new uniform random sampler for binary trees with $n$ internal nodes consuming $2n + \Theta(\log(n)^2)$ random bits on average. This makes it quasi-optimal and out-performs the classical Remy algorithm. We also present a sampler…

Since the works of Howard and Newman (2001), it is known that in straight radial rooted trees, with probability 1, infinite paths all have an asymptotic direction and each asymptotic direction is reached by (at least) an infinite path.…

The asyptotic number of nonequivalent binary n-codes is determined. This is also the asymptotic number of nonisomorphic binary n-matroids. The connection to a result of Lefmann, Roedl, Phelps is explored. The latter states that almost all…

We determine the sharp asymptotic scale of the probability that two uniformly random permutations are comparable in weak Bruhat order, showing that $\mathbb{P}(\sigma_1 \preceq_W \sigma_2)=\exp\Bigl(\bigl(-\tfrac12+o(1)\bigr)\,n\log…

We prove asymptotic normality for the number of fringe subtrees isomorphic to any given tree in uniformly random trees with given vertex degrees. As applications, we also prove corresponding results for random labelled trees with given…

Let $W_{\infty}(\beta)$ be the limit of the Biggins martingale $W_n(\beta)$ associated to a supercritical branching random walk with mean number of offspring $m$. We prove a functional central limit theorem stating that as $n\to\infty$ the…