arXiv subjects

Benoît Corsini

Publications and source records attributed to Benoît Corsini.

9 recordsLinked to original sources

arXiv2026

Minimal inversion of a permuton sample

Given a permutation $σ$, its corresponding \textit{inversion graph} is obtained by adding an edge between $i σ(j)$. The first results on random inversion graphs come from Acan and Pittel, who studied the connected threshold for a uniform permutation with fixed inversion number, and Bhattacharya and Mukherjee, who mostly focused on the degrees of the graph when the permutation is chosen uniformly at random. In this work, we call \textit{minimal inversion} the minimal degree of the inversion graph and extend a theorem from Bhattacharya and Mukherjee to the case where the permutation is not only uniform, but obtained as the ordering of points sampled according to some distribution on the plane. Under regularity assumptions on the distribution, and for the appropriate $α>0$, we show that the probability that the minimal inversion rescaled by $n^{α/(α+1)}$ is larger than $t$ behaves like $\exp(-ct^{α+1})$ for some constant $c>0$ depending on the distribution. We further show that every $α>0$ admits at least one corresponding distribution, thus proving that the minimal inversion can asymptotically scale as $n^β$ for any $β\in[0,1]$ (the cases $β=0$ and $β=1$ being obtained via the identity and anti-identity permutations, among others).

math.PR↗

arXiv2025

On exponentially height-penalized random trees

Given $n \in \mathbb{N}$ and $μ\in \mathbb{R}$, a $\textit{$μ$-height-biased tree of size $n$}$ is a random plane tree $\mathbf{\mathbf{T}}_n$ with $n$ vertices with law given by $\mathbb{P}(\mathbf{T}=t) \propto e^{-μh(t)}$, where $t$ ranges over fixed plane trees with $n$ vertices, and $h(t)$ is the height of $t$. Fix a sequence $(μ_n)_{n \ge 1}$ of real numbers, and for $n \ge 1$ let $\mathbf{T}_n$ be a $μ$-height-biased tree of size $n$. Durhuus and Ünel (2023) described the asymptotic behaviour of $h(\mathbf{T}_n)$ when $μ_n \equiv μ\in \mathbb{R}$ is fixed. In this work, we extend their results to arbitrary sequences of positive parameters depending on $n$. Most notably, we show that such a tree behaves like a height-biased Continuum Random Tree (CRT) when $μ_n$ is of order $1/\sqrt{n}$; that its height is asymptotically $(2π^2n/μ_n)^{1/3}$ when $μ_n$ is of larger order than $1/\sqrt{n}$ and of smaller order than $n$; and that its height converges to a fixed constant when $μ_n$ is of order at least $n$, with some random jumps under specific conditions on $μ_n$. We additionally prove various results on second order behaviours, and large deviation principles for the height, for different regimes of $μ_n$. Finally, we describe new statistics of these trees, covering their widths, their root degrees, and the local structure around their roots.

math.PR↗

arXiv2025

Local limit of Prim's algorithm

We study the local evolution of Prim's algorithm on large finite weighted graphs. When performed for $n$ steps, where $n$ is the size of the graph, Prim's algorithm will construct the minimal spanning tree (MST). We assume that our graphs converge locally in probability to some limiting rooted graph. In that case, Aldous and Steele already proved that the local limit of the MST converges to a limiting object, which can be thought of as the MST on the limiting infinite rooted graph. Our aim is to investigate {\em how} the local limit of the MST is reached \textit{dynamically}. For this, we take $tn+o(n)$ steps of Prim, for $t\in[0,1]$, and, under some reasonable assumptions, show how the local structure interpolates between performing Prim's algorithm on the local limit when $t=0$, to the full local limit of the MST for $t=1$. Our proof relies on the use of the recently developed theory of {\em dynamic local convergence}. We further present several examples for which our assumptions, and thus our results, apply.

math.PR↗

arXiv2025

Binary search trees of permuton samples

Binary search trees (BST) are a popular type of data structure when dealing with ordered data. Indeed, they enable one to access and modify data efficiently, with their height corresponding to the worst retrieval time. From a probabilistic point of view, binary search trees associated with data arriving in a uniform random order are well understood, but less is known when the input is a non-uniform random permutation. We consider here the case where the input comes from i.i.d. random points in the plane with law $μ$, a model which we refer to as a permuton sample. Our results show that the asymptotic proportion of nodes in each subtree depends on the behavior of the measure $μ$ at its left boundary, while the height of the BST has a universal asymptotic behavior for a large family of measures $μ$. Our approach involves a mix of combinatorial and probabilistic tools, namely combinatorial properties of binary search trees, coupling arguments, and deviation estimates.

math.PR↗

arXiv2023

Limits of Mallows trees

This article studies the limit of binary search trees drawn from Mallows permutations under various topologies. The main result, pertaining to the standard local topology for graphs, requires the introduction of a generalization of binary search trees to two-sided infinite sequences, referred to as \textit{redwood trees}. We then show that the almost-sure local limit of finite Mallows trees is the redwood tree drawn from the two-sided infinite Mallows permutation, thus corresponding to swapping the local limit and the binary search tree structure. Building off this result, we study various other natural topologies: the rooted topology of the local structure around the root, the Gromov-Hausdorff-Prokhorov topology of the tree seen as a metric space, and the subtree size topology of the ratio of nodes split between left and right subtrees. The limit of Mallows trees under these three topologies combined with the case of the local topology allow us to draw a global picture of what large Mallows trees look like from different perspective and further strengthen the relation between finite and infinite Mallows permutations.

math.PR↗

arXiv2022

Continuous-time Mallows processes

In this article, we introduce \textit{Mallows processes}, defined to be continuous-time càdlàg processes with Mallows distributed marginals. We show that such processes exist and that they can be restricted to have certain natural properties. In particular, we prove that there exists \textit{regular} Mallows processes, defined to have their inversions numbers $\mathrm{Inv}_j(σ)=|\{i\in[j-1]:σ(i)>σ(j)\}|$ be independent increasing stochastic processes with jumps of size $1$. We further show that there exists a unique Markov process which is a regular Mallows process. Finally, we study properties of regular Mallows processes and show various results on the structure of these objects. Among others, we prove that the graph structure related to regular Mallows processes looks like an \textit{expanded hypercube} where we stacked $k$ hypercubes on the dimension $k\in[n]$; we also prove that the first jumping times of regular Mallows processes converge to a Poisson point process.

math.PR↗

arXiv2022

Finding minimum spanning trees via local improvements

We consider a family of local search algorithms for the minimum-weight spanning tree, indexed by a parameter $ρ$. One step of the local search corresponds to replacing a connected induced subgraph of the current candidate graph whose total weight is at most $ρ$ by the minimum spanning tree (MST) on the same vertex set. Fix a non-negative random variable $X$, and consider this local search problem on the complete graph $K_n$ with independent $X$-distributed edge weights. Under rather weak conditions on the distribution of $X$, we determine a threshold value $ρ^*$ such that the following holds. If the starting graph (the "initial candidate MST") is independent of the edge weights, then if $ρ> ρ^*$ local search can construct the MST with high probability (tending to $1$ as $n \to \infty$), whereas if $ρ< ρ^*$ it cannot with high probability.

math.PR↗

arXiv2021

The height of record-biased trees

Given a permutation $σ$, its corresponding binary search tree is obtained by recursively inserting the values $σ(1),\ldots,σ(n)$ into a binary tree so that the label of each node is larger than the labels of its left subtree and smaller than the labels of its right subtree. In 1986, Devroye proved that the height of such trees when $σ$ is a random uniform permutation is of order $(c^*+o_\mathbb{P}(1))\log n$ as $n$ tends to infinity, where $c^*$ is the only solution to $c\log(2e/c)=1$ with $c\geq2$. In this paper, we study the height of binary search trees drawn from the record-biased model of permutations, introduced by Auger, Bouvel, Nicaud, and Pivoteau in 2016. The record-biased distribution is the probability measure on the set of permutations whose weight is proportional to $θ^{\mathrm{record}(σ)}$, where $\mathrm{record}(σ)=|\{i\in[n]:\forall j σ(j)\}|$. We show that the height of a binary search tree built from a record-biased permutation of size $n$ with parameter $θ$ is of order $(1+o_\mathbb{P}(1))\max\{c^*\log n,\,θ\log(1+n/θ)\}$, hence giving a full characterization of the first order asymptotic behaviour of the height of such trees.

math.PR↗

arXiv2020

The height of Mallows trees

Random binary search trees are obtained by recursively inserting the elements $σ(1),σ(2),\ldots,σ(n)$ of a uniformly random permutation $σ$ of $[n]=\{1,\dots,n\}$ into a binary search tree data structure. Devroye (1986) proved that the height of such trees is asymptotically of order $c^*\log n$, where $c^*=4.311\ldots$ is the unique solution of $c \log((2e)/c)=1$ with $c \geq 2$. In this paper, we study the structure of binary search trees $T_{n,q}$ built from Mallows permutations. A $\textrm{Mallows}(q)$ permutation is a random permutation of $[n]=\{1,\ldots,n\}$ whose probability is proportional to $q^{\textrm{Inv}(σ)}$, where $\textrm{Inv}(σ) = \#\{i < j: σ(i) > σ(j)\}$. This model generalizes random binary search trees, since $\textrm{Mallows}(q)$ permutations with $q=1$ are uniformly distributed. The laws of $T_{n,q}$ and $T_{n,q^{-1}}$ are related by a simple symmetry (switching the roles of the left and right children), so it suffices to restrict our attention to $q\leq1$. We show that, for $q\in[0,1]$, the height of $T_{n,q}$ is asymptotically $(1+o(1))(c^* \log n + n(1-q))$ in probability. This yields three regimes of behaviour for the height of $T_{n,q}$, depending on whether $n(1-q)/\log n$ tends to zero, tends to infinity, or remains bounded away from zero and infinity. In particular, when $n(1-q)/\log n$ tends to zero, the height of $T_{n,q}$ is asymptotically of order $c^*\log n$, like it is for random binary search trees. Finally, when $n(1-q)/\log n$ tends to infinity, we prove stronger tail bounds and distributional limit theorems for the height of $T_{n,q}$.

math.PR↗