arXiv subjects

Louigi Addario-Berry

Publications and source records attributed to Louigi Addario-Berry.

At least 37 records · Page 2Linked to original sources

arXiv2024

The top eigenvalue of uniformly random trees

Let ${\mathbf T}_n$ be a uniformly random tree with vertex set $[n]=\{1,\ldots,n\}$, let $Δ_{{\mathbf T}_n}$ be the largest vertex degree in ${\mathbf T}_n$, and let $λ_1({\mathbf T}_n),\ldots,λ_n({\mathbf T}_n)$ be the eigenvalues of its adjacency matrix, arranged in decreasing order. We prove that $|λ_1({\mathbf T}_n)-\sqrt{Δ_{{\mathbf T}_n}}| \to 0$ in expectation as $n \to \infty$, and additionally prove probability tail bounds for $|λ_1({\mathbf T}_n)-\sqrt{Δ_{{\mathbf T}_n}}|$. Writing $a_n$ for any median of $Δ_{{\mathbf T}_n}$, we also prove that $|λ_k({\mathbf T}_n)-\sqrt{a_n}| \to 0$ in expectation, uniformly over $1 \le k \le e^{\log^β(n)}$, for any fixed $β\in (0,1/2)$. The proof is based on the trace method and thus on counting closed walks in a random tree. To this end, we develop novel combinatorial tools for encoding walks in trees that we expect will find other applications. In order to apply these tools, we show that uniformly random trees -- after appropriate "surgery" -- satisfy, with high probability, the properties required for the combinatorial bounds to be effective.

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arXiv2024

Random trees have height $O(\sqrt{n})$

We obtain new non-asymptotic tail bounds for the height of uniformly random trees with a given degree sequence, simply generated trees and conditioned Bienaymé trees (the family trees of branching processes), in the process settling three conjectures of Janson (2012) and answering several other questions from the literature. Moreover, we define a partial ordering on degree sequences and show that it induces a stochastic ordering on the heights of uniformly random trees with given degree sequences. The latter result can also be used to show that sub-binary random trees are stochastically the tallest trees with a given number of vertices and leaves (and thus that random binary trees are the stochastically tallest random homeomorphically irreducible trees with a given number of vertices). Our proofs are based in part on the bijection between trees and sequences introduced by Foata and Fuchs (1970), which can be recast to provide a line-breaking construction of random trees with given vertex degrees as shown in Addario-Berry, Blanc-Renaudie, Donderwinkel, Maazoun and Martin (2023).

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arXiv2022

The central limit theorem via doubling of variables

We give a new, self-contained proof of the multidimensional central limit theorem using the technique of ``doubling variables," which is traditionally used to prove uniqueness of solutions of partial differential equations (PDEs). Our technique also yields quantitative bounds for random variables with finite $2+γ$ moment for some $γ\in (0,1]$; when $γ=1$, this proves a version of the Berry--Esseen theorem in $\mathbb{R}^d$.

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arXiv2022

The Foata-Fuchs proof of Cayley's formula, and its probabilistic uses

We present a very simple bijective proof of Cayley's formula due to Foata and Fuchs (1970). This bijection turns out to be very useful when seen through a probabilistic lens; we explain some of the ways in which it can be used to derive probabilistic identities, bounds, and growth procedures for random trees with given degrees, including random d-ary trees. We also introduce a partial order on the degree sequences of rooted trees, and conjecture that it induces a stochastic partial order on heights of random rooted trees with given degrees.

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arXiv2022

Multi-source invasion percolation on the complete graph

We consider invasion percolation on the randomly-weighted complete graph $K_n$, started from some number $k(n)$ of distinct source vertices. The outcome of the process is a forest consisting of $k(n)$ trees, each containing exactly one source. Let $M_n$ be the size of the largest tree in this forest. Logan, Molloy and Pralat (arXiv:1806.10975) proved that if $k(n)/n^{1/3} \to 0$ then $M_n/n \to 1$ in probability. In this paper we prove a complementary result: if $k(n)/n^{1/3} \to \infty$ then $M_n/n \to 0$ in probability. This establishes the existence of a phase transition in the structure of the invasion percolation forest around $k(n) \asymp n^{1/3}$. Our arguments rely on the connection between invasion percolation and critical percolation, and on a coupling between multi-source invasion percolation with differently-sized source sets. A substantial part of the proof is devoted to showing that, with high probability, a certain fragmentation process on large random binary trees leaves no components of macroscopic size.

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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.

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arXiv2022

Universal height and width bounds for random trees

We prove non-asymptotic stretched exponential tail bounds on the height of a randomly sampled node in a random combinatorial tree, which we use to prove bounds on the heights and widths of random trees from a variety of models. Our results allow us to prove a conjecture and settle an open problem of Janson (https://doi.org/10.1214/11-PS188), and nearly prove another conjecture and settle another open problem from the same work (up to a polylogarithmic factor). The key tool for our work is an equivalence in law between the degrees along the path to a random node in a random tree with given degree statistics, and a random truncation of a size-biased ordering of the degrees of such a tree. We also exploit a Poissonization trick introduced by Camarri and Pitman (https://doi.org/10.1214/EJP.v5-58) in the context of inhomogeneous continuum random trees, which we adapt to the setting of random trees with fixed degrees. Finally, we propose and justify a change to the conventions of branching process nomenclature: the name "Galton-Watson trees" should be permanently retired by the community, and replaced with the name "Bienaymé trees".

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arXiv2022

Symmetric cooperative motion in one dimension

We explore the relationship between recursive distributional equations and convergence results for finite difference schemes of parabolic partial differential equations (PDEs). We focus on a family of random processes called symmetric cooperative motions, which generalize the symmetric simple random walk and the symmetric hipster random walk introduced in [Addario-Berry, Cairns, Devroye, Kerriou and Mitchell, arXiv:1909.07367]. We obtain a distributional convergence result for symmetric cooperative motions and, along the way, obtain a novel proof of the Bernoulli central limit theorem. In addition, we prove a PDE result relating distributional solutions and viscosity solutions of the porous medium equation and the parabolic $p$-Laplace equation, respectively, in one dimension.

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arXiv2021

Asymmetric cooperative motion in one dimension

We prove distributional convergence for a family of random processes on $\mathbb{Z}$, which we call asymmetric cooperative motions. The model generalizes the "totally asymmetric hipster random walk" introduced in [Addario-Berry, Cairns, Devroye, Kerriou and Mitchell, 2020]. We present a novel approach based on connecting a temporal recurrence relation satisfied by the cumulative distribution functions of the process to the theory of finite difference schemes for Hamilton-Jacobi equations [Crandall and Lyons, 1984]. We also point out some surprising lattice effects that can persist in the distributional limit, and propose several generalizations and directions for future research.

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arXiv2021

Barycentric Brownian Bees

We establish an invariance principle for the barycenter of a Brunet-Derrida particle system in $d$ dimensions. The model consists of $N$ particles undergoing dyadic branching Brownian motion with rate $1$. At a branching event, the number of particles is kept equal to $N$ by removing the particle located furthest away from the barycenter. To prove the invariance principle, a key step is to establish Harris recurrence for the process viewed from its barycenter.

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arXiv2021

Geometry of the minimal spanning tree of a random $3$-regular graph

The global structure of the minimal spanning tree (MST) is expected to be universal for a large class of underlying random discrete structures. However, very little is known about the intrinsic geometry of MSTs of most standard models, and so far the scaling limit of the MST viewed as a metric measure space has only been identified in the case of the complete graph [5]. In this work, we show that the MST constructed by assigning i.i.d. continuous edge-weights to either the random (simple) $3$-regular graph or the $3$-regular configuration model on $n$ vertices, endowed with the tree distance scaled by $n^{-1/3}$ and the uniform probability measure on the vertices, converges in distribution with respect to Gromov-Hausdorff-Prokhorov topology to a random compact metric measure space. Further, this limiting space has the same law as the scaling limit of the MST of the complete graph identified in [5] up to a scaling factor of $6^{1/3}$. Our proof relies on a novel argument that proceeds via a comparison between a $3$-regular configuration model and the largest component in the critical Erdős-Rényi random graph. The techniques of this paper can be used to establish the scaling limit of the MST in the setting of general random graphs with given degree sequences provided two additional technical conditions are verified.

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arXiv2021

Broadcasting on random recursive trees

We study the broadcasting problem when the underlying tree is a random recursive tree. The root of the tree has a random bit value assigned. Every other vertex has the same bit value as its parent with probability $1-q$ and the opposite value with probability $q$, where $q \in [0,1]$. The broadcasting problem consists in estimating the value of the root bit upon observing the unlabeled tree, together with the bit value associated with every vertex. In a more difficult version of the problem, the unlabeled tree is observed but only the bit values of the leaves are observed. When the underlying tree is a uniform random recursive tree, in both variants of the problem we characterize the values of $q$ for which the optimal reconstruction method has a probability of error bounded away from $1/2$. We also show that the probability of error is bounded by a constant times $q$. Two simple reconstruction rules are analyzed in detail. One of them is the simple majority vote, the other is the bit value of the centroid of the tree. Most results are extended to linear preferential attachment trees as well.

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arXiv2021

Random tree-weighted graphs

For each $n \ge 1$, let $\mathrm{d}^n=(d^{n}(i),1 \le i \le n)$ be a sequence of positive integers with even sum $\sum_{i=1}^n d^n(i) \ge 2n$. Let $(G_n,T_n,Γ_n)$ be uniformly distributed over the set of simple graphs $G_n$ with degree sequence $\mathrm{d}^n$, endowed with a spanning tree $T_n$ and rooted along an oriented edge $Γ_n$ of $G_n$ which is not an edge of $T_n$. Under a finite variance assumption on degrees in $G_n$, we show that, after rescaling, $T_n$ converges in distribution to the Brownian continuum random tree as $n \to \infty$. Our main tool is a new version of Pitman's additive coalescent (https://doi.org/10.1006/jcta.1998.2919), which can be used to build both random trees with a fixed degree sequence, and random tree-weighted graphs with a fixed degree sequence. As an input to the proof, we also derive a Poisson approximation theorem for the number of loops and multiple edges in the superposition of a fixed graph and a random graph with a given degree sequence sampled according to the configuration model; we find this to be of independent interest.

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arXiv2020

Convergence of odd-angulations via symmetrization of labeled trees

Fix $p\geq 5$ an odd integer integer. Let $M_n$ be a uniform $p$-angulation with $n$ vertices and endowed with the uniform probability measure on its vertices. We prove that, there exists $C_p\in \mathbb{R}_+$ such that, after rescaling distances by $C_p/n^{1/4}$, $M_n$ converges in distribution for the Gromov-Hausdorff-Prokhorov topology towards the Brownian map. To prove the preceding fact, we introduce a `bootstrapping' principle for distributional convergence of random labelled plane trees. In particular, the latter allows to obtain an invariance principle for labeled multitype Galton-Watson trees, with only a weak assumption on the centering of label displacements

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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 $T_{n,q}$.

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arXiv2020

A probabilistic approach to the leader problem in random graphs

We study the fixation time of the identity of the leader, i.e., the most massive component, in the general setting of Aldous's multiplicative coalescent [4, 5], which in an asymptotic sense describes the evolution of the component sizes of a wide array of near-critical coalescent processes, including the classical Erdős-Rényi process. We show tightness of the fixation time in the "Brownian" regime, explicitly determining the median value of the fixation time to within an optimal $O(1)$ window. This generalizes Łuczak's result [31] for the Erdős-Rényi random graph using completely different techniques. In the heavy-tailed case, in which the limit of the component sizes can be encoded using a thinned pure-jump Lévy process, we prove that only one-sided tightness holds. This shows a genuine difference in the possible behavior in the two regimes. The solution to the leader problem in the setting of the Erdős-Rényi random graph played an important role in the study of the scaling limit of the minimal spanning tree on the complete graph [2]. We believe that analogous results, such as those proved herein, will be useful in establishing universality of the intrinsic geometry of the minimal spanning tree across a large class of models.

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arXiv2019

Hipster random walks

We introduce and study a family of random processes on trees we call hipster random walks, special instances of which we heuristically connect to the min-plus binary trees introduced by Robin Pemantle and studied by Auffinger and Cable (2017; arXiv:1709.07849), and to the critical random hierarchical lattice studied by Hambly and Jordan (2004). We prove distributional convergence for the processes by showing that their evolutions can be understood as a discrete analogues of certain convection-diffusion equations, then using a combination of coupling arguments and results from the numerical analysis literature on convergence of numerical approximations of PDEs.

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arXiv2019

The algorithmic hardness threshold for continuous random energy models

We prove an algorithmic hardness result for finding low-energy states in the so-called \emph{continuous random energy model (CREM)}, introduced by Bovier and Kurkova in 2004 as an extension of Derrida's \emph{generalized random energy model}. The CREM is a model of a random energy landscape $(X_v)_{v \in \{0,1\}^N}$ on the discrete hypercube with built-in hierarchical structure, and can be regarded as a toy model for strongly correlated random energy landscapes such as the family of $p$-spin models including the Sherrington--Kirkpatrick model. The CREM is parameterized by an increasing function $A:[0,1]\to[0,1]$, which encodes the correlations between states. We exhibit an \emph{algorithmic hardness threshold} $x_*$, which is explicit in terms of $A$. More precisely, we obtain two results: First, we show that a renormalization procedure combined with a greedy search yields for any $\varepsilon > 0$ a linear-time algorithm which finds states $v \in \{0,1\}^N$ with $X_v \ge (x_*-\varepsilon) N$. Second, we show that the value $x_*$ is essentially best-possible: for any $\varepsilon > 0$, any algorithm which finds states $v$ with $X_v \ge (x_*+\varepsilon)N$ requires exponentially many queries in expectation and with high probability. We further discuss what insights this study yields for understanding algorithmic hardness thresholds for random instances of combinatorial optimization problems.

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