Related papers: Exploring Werner Krandick's Binary Tree Jump Stati…
We use statistical estimates of the entropy rate of spike train data in order to make inferences about the underlying structure of the spike train itself. We first examine a number of different parametric and nonparametric estimators (some…
Random forests are ensemble methods which grow trees as base learners and combine their predictions by averaging. Random forests are known for their good practical performance, particularly in high dimensional set-tings. On the theoretical…
The simple random walk on $\mathbb{Z}^p$ shows two drastically different behaviours depending on the value of $p$: it is recurrent when $p\in\{1,2\}$ while it escapes (with a rate increasing with $p$) as soon as $p\geq3$. This classical…
When we enumerate numbers up to some specific value, or, even if we do not specify the number, we know at the same time that there are much greater numbers which should be reachable by the same enumeration, but indeed we also congnize them…
Consider a rooted binary tree with n nodes. Assign with the root the abscissa 0, and with the left (resp. right) child of a node of abscissa i the abscissa i-1 (resp. i+1). We prove that the number of binary trees of size n having exactly…
For a two-dimensional lattice $\Lambda$ with $n$ vertices, the number of spanning trees $N_{ST}(\Lambda)$ grows asymptotically as $\exp(n z_\Lambda)$ in the thermodynamic limit. We present exact integral expression and numerical value for…
Weighted recursive trees are built by adding successively vertices with predetermined weights to a tree: each new vertex is attached to a parent chosen randomly proportionally to its weight. Under some assumptions on the sequence of…
A jump is a pair of consecutive elements in an extension of a poset which are incomparable in the original poset. The arboreal jump number is an NP-hard problem that aims to find an arboreal extension of a given poset with minimum number of…
We use reverse mathematics to analyze "iterated jump" versions of the following four principles: the atomic model theorem with subenumerable types (AST), the diagonally noncomputable principle (DNR), weak weak K\H{o}nig's lemma (WWKL), and…
The k-Tree algorithm [Wagner 02] is a non-trivial algorithm for the average-case k-SUM problem that has found widespread use in cryptanalysis. Its input consists of k lists, each containing n integers from a range of size m. Wagner's…
Signs of hierarchy are prevalent in a wide range of systems in nature and society. One of the key problems is quantifying the importance of hierarchical organisation in the structure of the network representing the interactions or…
We consider the number distribution of topological defects resulting from the finite-time crossing of a continuous phase transition and identify signatures of universality beyond the mean value, predicted by the Kibble-Zurek mechanism.…
Due to their long-standing reputation as excellent off-the-shelf predictors, random forests continue remain a go-to model of choice for applied statisticians and data scientists. Despite their widespread use, however, until recently, little…
A continuously measured quantum system with multiple jump channels gives rise to a stochastic process described by random jump times and random emitted symbols, representing each jump channel. While much is known about the waiting time…
We study here a very popular 1D jump model introduced by Kac: it consists of $N$ velocities encountering random binary collisions at which they randomly exchange energy. We show the uniform (in $N$) exponential contractivity of the dynamics…
The most fundamental problem in statistical causality is determining causal relationships from limited data. Probability trees, which combine prior causal structures with Bayesian updates, have been suggested as a possible solution. In this…
It has been observed by Belkin et al.\ that over-parametrized neural networks exhibit a `double descent' phenomenon. That is, as the model complexity (as reflected in the number of features) increases, the test error initially decreases,…
We investigate the statistics of trees grown from some initial tree by attaching links to preexisting vertices, with attachment probabilities depending only on the valence of these vertices. We consider the asymptotic mass distribution that…
Parking problems derive from works in combinatorics by Konheim and Weiss in the 1960s. In a memorable contribution, Lackner and Panholzer (2016) studied parking on a random tree and established a phase transition for this process when \(m…
A novel version of the Continuous-Time Random Walk (CTRW) model with memory is developed. This memory means the dependence between arbitrary number of successive jumps of the process, while waiting times between jumps are considered as…