相关论文: Congruence properties of depths in some random tre…

We give sufficient conditions under which a random graph with a specified degree sequence is symmetric or asymmetric. In the case of bounded degree sequences, our characterisation captures the phase transition of the symmetry of the random…

We study the fundamental question of how likely it is that two randomly chosen trees are isomorphic to each other for different models of random trees. We show that the probability decays exponentially for rooted labeled trees as well as…

Let $a_1 = 1$ and, for $n > 1$, $a_n = a_{n-1} + a_{\left \lfloor \frac{n}{2} \right \rfloor}$. In this paper we will look at congruence properties and the growth rate of this sequence. First we will show that if $x \in \{1, 2, 3, 5, 6, 7…

We show that for any fixed dense graph G and bounded-degree tree T on the same number of vertices, a modest random perturbation of G will typically contain a copy of T . This combines the viewpoints of the well-studied problems of embedding…

We study the asymptotic behavior of ``true" self-avoiding random walks on general infinite locally finite trees. In this model, the walk starts at the root and, at each step, from its current vertex chooses a neighboring edge to traverse…

Uncover the vertices of a given graph, deterministic or random, in random order; we consider both a discrete-time and a continuous-time version. We study the evolution of the number of visible edges, and show convergence after normalization…

We provide asymptotics for the range R(n) of a random walk on the d-dimensional lattice indexed by a random tree with n vertices. Using Kingman's subadditive ergodic theorem, we prove under general assumptions that R(n)/n converges to a…

We study the random m-ary search tree model (where m stands for the number of branches of a search tree), an important problem for data storage in computer science, using a variety of statistical physics techniques that allow us to obtain…

We study various types of consistency of honest decision trees and random forests in the regression setting. In contrast to related literature, our proofs are elementary and follow the classical arguments used for smoothing methods. Under…

We consider random graphs with a given degree sequence and show, under weak technical conditions, asymptotic normality of the number of components isomorphic to a given tree, first for the random multigraph given by the configuration model…

Consider a rooted infinite Galton-Watson tree with mean offspring number $m>1$, and a collection of i.i.d. positive random variables $\xi_e$ indexed by all the edges in the tree. We assign the resistance $m^d \xi_e$ to each edge $e$ at…

Given any regularly varying dislocation measure, we identify a natural self-similar fragmentation tree as scaling limit of discrete fragmentation trees with unit edge lengths. As an application, we obtain continuum random tree limits of…

Asymptotics of the variances of many cost measures in random digital search trees are often notoriously messy and involved to obtain. A new approach is proposed to facilitate such an analysis for several shape parameters on random symmetric…

The probability that two randomly selected phylogenetic trees of the same size are isomorphic is found to be asymptotic to a decreasing exponential modulated by a polynomial factor. The number of symmetrical nodes in a random phylogenetic…

We study the evolution of graphs densifying by adding edges: Two vertices are chosen randomly, and an edge is (i) established if each vertex belongs to a tree; (ii) established with probability $p$ if only one vertex belongs to a tree;…

We give new general formulas for the asymptotics of the number of spanning trees of a large graph. A special case answers a question of McKay (1983) for regular graphs. The general answer involves a quantity for infinite graphs that we call…

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…

Despite recent theoretical progress on the non-convex optimization of two-layer neural networks, it is still an open question whether gradient descent on neural networks without unnatural modifications can achieve better sample complexity…

Numerous temporal inference tasks such as fault monitoring and anomaly detection exhibit a persistence property: for example, if something breaks, it stays broken until an intervention. When modeled as a Dynamic Bayesian Network,…

The space requirements of an $m$-ary search tree satisfies a well-known phase transition: when $m\leq 26$, the second order asymptotics is Gaussian. When $m\geq 27$, it is not Gaussian any longer and a limit $W$ of a complex-valued…