Related papers: On the Asymptotic Behavior of Guessing Sequences
The exponential growth rate of non polynomially growing subgroups of $GL_d$ is conjectured to admit a uniform lower bound. This is known for non-amenable subgroups, while for amenable subgroups it is known to imply the Lehmer conjecture…
We use various combinatorial and probabilistic techniques to study growth rates for the probability that a random permutation from the Mallows distribution avoids consecutive patterns. The Mallows distribution behaves like a $q$-analogue of…
Given a hypergraph $\Gamma=(\Omega,\mathcal{X})$ and a sequence $\mathbf{p} = (p_\omega)_{\omega\in \Omega}$ of values in $(0,1)$, let $\Omega_{\mathbf{p}}$ be the random subset of $\Omega$ obtained by keeping every vertex $\omega$…
Given a sequence of complex square matrices, $a_n$, consider the sequence of their partial products, defined by $p_n=p_{n-1}a_{n}$. What can be said about the asymptotics as $n\to\infty$ of the sequence $f(p_n)$, where $f$ is a continuous…
Computing the reachability probability in infinite state probabilistic models has been the topic of numerous works. Here we introduce a new property called \emph{divergence} that when satisfied allows to compute reachability probabilities…
An \emph{evolving Shelah-Spencer process} is one by which a random graph grows, with at each time $\tau \in {\bf N}$ a new node incorporated and attached to each previous node with probability $\tau^{-\alpha}$, where $\alpha \in (0,1)…
Let $\mm_n, n=0,1,...$ be the supercritical branching random walk, in which the number of direct descendants of one individual may be infinite with positive probability. Assume that the standard martingale $W_n$ related to $\mm_n$ is…
In this monograph, we prove an asymptotic approximation for integrals of probability densities over sets in finite dimensional euclidean space, which are far away from the origin (asymptotic sets). We use this approximation to investigate…
We address the question of the infinitude of twin and cousin prime pairs from a probabilistic perspective. Our approach partitions the set of integer numbers greater than $2$ in finite intervals of the form $[p_{n-1}^2,p_n^2)$, $p_{n-1}$…
We introduce asymptotic R\'enyi entropies as a parameterized family of invariants for random walks on groups. These invariants interpolate between various well-studied properties of the random walk, including the growth rate of the group,…
We consider the biased random walk on a tree constructed from the set of finite self-avoiding walks on a lattice, and use it to construct probability measures on infinite self-avoiding walks. The limit measure (if it exists) obtained when…
In this paper, we are interested in the asymptotic behaviour of the sequence of processes $(W_n(s,t))_{s,t\in[0,1]}$ with \begin{equation*} W_n(s,t):=\sum_{k=1}^{\lfloor nt\rfloor}\big(1_{\{\xi_{S_k}\leq s\}}-s\big) \end{equation*} where…
We consider a model of random tree growth, where at each time unit a new vertex is added and attached to an already existing vertex chosen at random. The probability with which a vertex with degree $k$ is chosen is proportional to $w(k)$,…
In random walk theory, it is customary to assume that a given walk is irreducible and/or aperiodic. While these prevailing assumptions make particularly tractable the analysis of random walks and help to highlight their diffusive nature,…
This is the second in a series of articles devoted to showing that a typical covering map of large degree to a fixed, regular graph has its new adjacency eigenvalues within the bound conjectured by Alon for random regular graphs. The first…
The last decade has witnessed a growing interest in random forest models which are recognized to exhibit good practical performance, especially in high-dimensional settings. On the theoretical side, however, their predictive power remains…
Random forests remain among the most popular off-the-shelf supervised learning algorithms. Despite their well-documented empirical success, however, until recently, few theoretical results were available to describe their performance and…
This paper studies higher index theory for a random sequence of bounded degree, finite graphs with diameter tending to infinity. We show that in a natural model for such random sequences the following hold almost surely: the coarse…
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…
For a permuton $\mu$ let $H_n(\mu)$ denote the Shannon entropy of the sampling distribution of $\mu$ on $n$ points. We investigate the asymptotic growth of $H_n(\mu)$ for a wide class of permutons. We prove that if $\mu$ has a non-vanishing…