arXiv subjects

Robin Khanfir

Publications and source records attributed to Robin Khanfir.

13 recordsLinked to original sources

arXiv2026

Scaling limit of the range of tree-valued branching random walks in random environmen

We study a branching random walk (BRW) taking its values in a random tree $\bT$ (seen as a family tree) with an infinite line of ancestors that is a variant of a supercritical Galton--Watson (GW) tree with offspring distribution $\nu$. The transition probabilities of the BRW are those of a critical biased random walk on $\bT$: namely, the probability to move from $x$ to one of its $k_x$ children is $1/(\mathtt{m}_\nu+k_x)$ and the probability to move from $x$ to the direct parent of $x$ is $\mathtt{m}_\nu/(\mathtt{m}_\nu+k_x)$. Here $\ttm_\nu$ stands for the mean of $\nu$. The BRW is indexed by a critical GW tree conditioned to have $n$ {vertices} and whose offspring distribution is in the domain of attraction of an $\alpha$-stable law with $\alpha \ino (1, 2]$. We denote by $\cR_n$ the range of the BRW, i.e., ~the set of all sites in $\bT$ visited by the BRW. Under a moment assumption for $\nu$, we prove that if we view $\cR_n$ as a random subtree of $\bT$ equipped with its graph distance $d_{\mathtt{gr}}$ and with its occupation measure $\ttm^{_{(n)}}_{{\mathtt{occ}}}$ then there exists a scaling sequence $s_n \! \to \! \infty$ such that conditionally given the environment $\bT$, the measured metric space $(\cR_n, s_n^{-1}d_{\mathtt{gr}} , \frac{_1}{^n}\ttm^{_{(n)}}_{{\mathtt{occ}}} )$ weakly converges in the Gromov--Hausdorff--Prokhorov sense to a random measured compact real tree introduced by Curien, Le Gall \& Miermont in \cite{CuLGMi13} called the Brownian cactus with $\alpha$-stable branching mechanism. This work extends in random environment the result from D., K., Lin \& Torri \cite{DuKhLiTo22} which deals with the case where $\bT$ is a regular tree.

math.PR↗

arXiv2025

The largest common subtree of two random trees

We study the size and structure of the largest common subtree (LCS) between two independent Bienaym\'e trees conditioned to have size $n$. When the trees are critical with finite $2$nd and $(2+\kappa)$th moment respectively for some $\kappa>0$, we prove that the LCS has size of order $\sqrt{n}$, and is approximated by the length of three paths meeting at a central node. Moreover, we show that the largest common subtree between two critical independent Bienaym\'e trees with size $n$ and finite second moments may be much larger than $\sqrt{n}$, implying that our result is tight. We also pose a number of open questions and suggestions for future research.

math.PR↗

arXiv2025

Flimsy Spaces

We study $n$-flimsy spaces, which are the topological spaces that remain connected when removing fewer than $n$ points but become disconnected when removing exactly $n$ points. We show that no such space exists for $n \geq 3$, and that the compact $2$-flimsy spaces are precisely the dense and order-complete cyclically ordered sets equipped with their order topology. Furthermore, we examine variants of the definition obtained by replacing connectedness by path-connectedness, where paths are either parametrized by $[0,1]$ or by arbitrary compact linear continua.

math.GN↗

arXiv2025

Does freezing impede the growth of random recursive trees?

Uniform attachment with freezing is an extension of the classical model of random recursive trees, in which trees are recursively built by attaching new vertices to old ones. In the model of uniform attachment with freezing, vertices are allowed to freeze, in the sense that new vertices cannot be attached to already frozen ones. We study the impact of removing attachment and/or freezing steps on the height of the trees. We show in particular that removing an attachment step can increase the expected height, and that freezing cannot substantially decrease the height of random recursive trees. Our methods are based on coupling arguments.

math.PR↗

arXiv2024

Tight universal bounds on the height times the width of random trees

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 size $n$, this product is $O(n \log n)$ in probability, answering a question by Addario-Berry (2019). The order of this bound is tight in this generality.

math.PR↗

arXiv2024

Optimal root recovery for uniform attachment trees and $d$-regular growing trees

We consider root-finding algorithms for random rooted trees grown by uniform attachment. Given an unlabeled copy of the tree and a target accuracy $\varepsilon > 0$, such an algorithm outputs a set of nodes that contains the root with probability at least $1 - \varepsilon$. We focus on the algorithm introduced by Bubeck, Devroye and Lugosi (2017) and proved to be optimal by Crane and Xu (2021). We prove that, for the optimal algorithm, an output set of size $\exp(O(\log^{1/2}(1/\varepsilon)))$ suffices; this bound is sharp and answers a question of Bubeck, Devroye and Lugosi (2017). We prove similar bounds for random regular trees that grow by uniform attachment, strengthening a result of Khim and Loh (2017).

cs.DS↗

arXiv2024

Refined Horton-Strahler numbers I: a discrete bijection

The Horton-Strahler number of a rooted tree $T$ is the height of the tallest complete binary tree that can be homeomorphically embedded in $T$. The number of full binary trees with $n$ internal vertices and Horton-Strahler number $s$ is known to be the same as the number of Dyck paths of length $2n$ whose height $h$ satisfies $\lfloor \log_2(1+h)\rfloor=s$. In this paper, we present a new bijective proof of the above result, that in fact strengthens and refines it as follows. We introduce a sequence of trees $(\tau_i,i \ge 0)$ which "interpolates" the complete binary trees, in the sense that $\tau_{2^h-1}$ is the complete binary tree of height $h$ for all $h \ge 0$, and $\tau_{i+1}$ strictly contains $\tau_i$ for all $i \ge 0$. Defining $\mathcal{S}(T)$ to be the largest $i$ for which $\tau_i$ can be homeomorphically embedded in $T$, we then show that the number of full binary trees $T$ with $n$ internal vertices and with $\mathcal{S}(T)=h$ is the same as the number of Dyck paths of length $2n$ with height $h$. (We call $\mathcal{S}(T)$ the refined Horton-Strahler number of $T$.) Our proof is bijective and relies on a recursive decomposition of binary trees (resp. Dyck paths) into subtrees with strictly smaller refined Horton-Strahler number (resp. subpaths with strictly smaller height). In a subsequent paper, we will show that the bijection has a continuum analogue, which transforms a Brownian continuum random tree into a Brownian excursion and under which (a continuous analogue of) the refined Horton-Strahler number of the tree becomes the height of the excursion.

math.CO↗

arXiv2024

Fluctuations of the Horton-Strahler number of stable Galton-Watson trees

The Horton-Strahler number -- also called the register function -- is a combinatorial tool that quantifies the branching complexity of a rooted tree. We study the law of the Horton-Strahler number of stable Galton-Watson trees conditioned to have size $n$ (including the Catalan trees), which are the finite-dimensional marginals of stable L\'evy trees. While these random variables are known to grow as a multiple of $\ln n$ in probability, their fluctuations are not well understood because they are coupled with deterministic oscillations. To rule out the latter, we introduce a real-valued variant of the Horton-Strahler number. We show that a rescaled exponential of this quantity jointly converges in distribution to a measurable function of the scaling limit of the trees, i.e. the stable L\'evy tree. We call this limit the Strahler dilation and we discuss its similarities with the Horton-Strahler number.

math.PR↗

arXiv2023

The Horton-Strahler number of Galton-Watson trees with possibly infinite variance

The Horton-Strahler number, also known as the register function, provides a tool for quantifying the branching complexity of a rooted tree. We consider the Horton-Strahler number of critical Galton-Watson trees conditioned to have size $n$ and whose offspring distribution is in the domain of attraction of an $\alpha$-stable law with $\alpha\in [1, 2]$. We give tail estimates and when $\alpha\neq 1$, we prove that it grows as $\frac{1}{\alpha}\log_{\alpha/(\alpha-1)} n$ in probability. This extends the result in Brandenberger, Devroye \& Reddad [6] dealing with the finite variance case for which $\alpha=2$. We also characterize the cases where $\alpha=1$, namely the spectrally positive Cauchy regime, which exhibits more complex behaviors. Our proofs are new and probabilistic; they relate the Horton-Strahler number with other shape parameters such as the height or largest degree.

math.PR↗

arXiv2022

Study of a division-like property

We introduce a weak division-like property for noncommutative rings: a nontrivial ring is fadelian if for all nonzero $a,x$ there exist $b,c$ such that $x=ab+ca$. We prove properties of fadelian rings, and construct examples of such rings which are not division rings, as well as non-Noetherian and non-Ore examples. We have also formalized some of these results in the Lean proof assistant.

math.RA↗

arXiv2022

Convergences of looptrees coded by excursions

In order to study convergences of looptrees, we construct continuum trees and looptrees from real-valued c\`adl\`ag functions without negative jumps called excursions. We then provide a toolbox to manipulate the two resulting codings of metric spaces by excursions and we formalize the principle that jumps correspond to loops and that continuous growths correspond to branches. Combining these codings creates new metric spaces from excursions that we call vernation trees. They consist of a collection of loops and trees glued along a tree structure so that they unify trees and looptrees. We also propose a topological definition for vernation trees, which yields what we argue to be the right space to study convergences of looptrees. However, those first codings lack some functional continuity, so we adjust them. We thus obtain several limit theorems. Finally, we present some probabilistic applications, such as proving an invariance principle for random discrete looptrees.

math.PR↗

arXiv2021

Scaling limits of tree-valued branching random walks

We consider a branching random walk (BRW) taking its values in the $\mathtt{b}$-ary rooted tree $\mathbb W_{ \mathtt{b}}$ (i.e. the set of finite words written in the alphabet $\{ 1, \ldots, \mathtt{b} \}$, with $\mathtt{b}\! \geq \! 2$). The BRW is indexed by a critical Galton--Watson tree conditioned to have $n$ vertices; its offspring distribution is aperiodic and is in the domain of attraction of a $\gamma$-stable law, $\gamma \in (1, 2]$. The jumps of the BRW are those of a nearest-neighbour null-recurrent random walk on $\mathbb W_{ \mathtt{b}}$ (reflection at the root of $\mathbb W_{ \mathtt{b}}$ and otherwise: probability $1/2$ to move closer to the root of $\mathbb W_{ \mathtt{b}}$ and probability $1/(2\mathtt{b})$ to move away from it to one of the $\mathtt{b}$ sites above). We denote by $\mathcal R_{\mathtt{b}} (n)$ the range of the BRW in $\mathbb W_{ \mathtt{b}}$ which is the set of all sites in $\mathbb W_{\mathtt{b}}$ visited by the BRW. We first prove a law of large numbers for $\# \mathcal R_{\mathtt{b}} (n)$ and we also prove that if we equip $\mathcal R_{\mathtt{b}} (n)$ (which is a random subtree of $\mathbb W_{\mathtt{b}}$) with its graph-distance $d_{\mathtt{gr}}$, then there exists a scaling sequence $(a_n)_{n\in \mathbb N}$ satisfying $a_n \! \rightarrow \! \infty$ such that the metric space $(\mathcal R_{\mathtt{b}} (n), a_n^{-1}d_{\mathtt{gr}})$, equipped with its normalised empirical measure, converges to the reflected Brownian cactus with $\gamma$-stable branching mechanism: namely, a random compact real tree that is a variant of the Brownian cactus introduced by N. Curien, J-F. Le Gall and G. Miermont.

math.PR↗

arXiv2018

Time and place of the maximum for one-dimensional diffusion bridges and meanders

For three constrained Brownian motions, the excursion, the meander, and the reflected bridge, the densities of the maximum and of the time to reach it were expressed as double series by Majumdar, Randon-Furling, Kearney, and Yor (2008). Some of these series were regularized by Abel summation. Similar results for Bessel processes were obtained by Schehr and Le Doussal (2010) using the real space renormalization group method. Here this work is reviewed, and extended from the point of view of one-dimensional diffusion theory to some other diffusion processes including skew Brownian bridges and generalized Bessel meanders. We discuss the limits of the application of this method for other diffusion processes.

math.PR↗