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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., Comment: 12 pages, 5 figures

Let ${\mathbf T}_n$ be a uniformly random tree with vertex set $[n]=\{1,\ldots,n\}$, let $\Delta_{{\mathbf T}_n}$ be the largest vertex degree in ${\mathbf T}_n$, and let $\lambda_1({\mathbf T}_n),\ldots,\lambda_n({\mathbf T}_n)$ be the eigenvalues of its adjacency matrix, arranged in decreasing order. We prove that $|\lambda_1({\mathbf T}_n)-\sqrt{\Delta_{{\mathbf T}_n}}| \to 0$ in expectation as $n \to \infty$, and additionally prove probability tail bounds for $|\lambda_1({\mathbf T}_n)-\sqrt{\Delta_{{\mathbf T}_n}}|$. Writing $a_n$ for any median of $\Delta_{{\mathbf T}_n}$, we also prove that $|\lambda_k({\mathbf T}_n)-\sqrt{a_n}| \to 0$ in expectation, uniformly over $1 \le k \le e^{\log^\beta(n)}$, for any fixed $\beta \in (0,1/2)$. 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., Comment: 40 pages, 4 figures

The height of a random PATRICIA tree built from independent, identically distributed infinite binary strings with arbitrary diffuse probability distribution $\mu$ on $\{0,1\}^\mathbb{N}$ is studied. We show that the expected height grows asymptotically sublinearly in the number of leaves for any such $\mu$, but can be made to exceed any specific sublinear growth rate by choosing $\mu$ appropriately., Comment: Revised version. Accepted for publication in the proceedings of the 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA2024)

We establish lower tail bounds for the height, and upper tail bounds for the width, of critical size-conditioned Bienaym\'e trees. Our bounds are optimal at this level of generality. We also obtain precise asymptotics for offspring distributions within the domain of attraction of a Cauchy distribution, under a local regularity assumption. Finally, we pose some questions on the possible asymptotic behaviours of the height and width of critical size-conditioned Bienaym\'e trees., Comment: 30 pages

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 $2+\gamma$ moment for some $\gamma \in (0,1]$; when $\gamma=1$, this proves a version of the Berry--Esseen theorem in $\mathbb{R}^d$., Comment: 15 pages

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., Comment: 35 pages

We consider a family of local search algorithms for the minimum-weight spanning tree, indexed by a parameter $\rho$. One step of the local search corresponds to replacing a connected induced subgraph of the current candidate graph whose total weight is at most $\rho$ 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 $\rho^*$ such that the following holds. If the starting graph (the "initial candidate MST") is independent of the edge weights, then if $\rho > \rho^*$ local search can construct the MST with high probability (tending to $1$ as $n \to \infty$), whereas if $\rho < \rho^*$ it cannot with high probability., Comment: 24 pages

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., Comment: 33 pages, 0 figures

We obtain new non-asymptotic tail bounds for the height of uniformly random trees with a given degree sequence, simply generated trees and conditioned Bienaym\'e 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)., Comment: 43 pages, 6 figures

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., Comment: 11 pages, 2 figures

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 "Bienaym\'e trees".

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., Comment: 28 pages, to appear in Transactions of the AMS

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 $(G_n,T_n,\Gamma_n)$ 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 $\Gamma_n$ 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., Comment: 38 pages; version 2 has several minor corrections pointed out by a referee, and also corrects the proof of Corollary 4.2, which previously failed to address the case that $G_n$ has $(1+o(1))n$ edges

Random binary search trees are obtained by recursively inserting the elements $\sigma(1),\sigma(2),\ldots,\sigma(n)$ of a uniformly random permutation $\sigma$ 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 $q^{\textrm{Inv}(\sigma)}$, where $\textrm{Inv}(\sigma) = \#\{i < j: \sigma(i) > \sigma(j)\}$. 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 $T_{n,q}$., Comment: 52 pages

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.

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., Comment: 36 pages. This version incorporates useful feedback from an anonymous referee. To appear in Annals of Applied Probability

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., Comment: 28 pages

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, Comment: 25 pages

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., Comment: 22 pages, 2 figures. Minor additions and modifications in v2, minor corrections in v3 to v5, to appear in Mathematical Statistics and Learning

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 Erd\H{o}s-R\'enyi 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., Comment: 61 pages; to appear in Probability Theory and Related Fields

In this work we study a non-local version of the Fisher-KPP equation, \begin{equation*} \begin{cases} \frac{\partial u}{\partial t}=\tfrac{1}{2}\Delta u +u (1- \phi \ast u), \quad t>0, \quad x\in \mathbb{R}, u(0,x)=u_0(x), \quad x\in \mathbb{R} \end{cases} \end{equation*} and its relation to $\textit{branching Brownian motion with decay of mass}$ as introduced by Addario-Berry and Penington (2017), i.e. a particle system consisting of a standard branching Brownian motion (BBM) with a competitive interaction between nearby particles. Particles in the BBM with decay of mass have a position in $\mathbb{R}$ and a mass, branch at rate 1 into two daughter particles of the same mass and position, and move independently as Brownian motions. Particles lose mass at a rate proportional to the mass in a neighbourhood around them (as measured by the function $\phi$). We obtain two types of results. First, we study the behaviour of solutions to the partial differential equation above. We show that, under suitable conditions on $\phi$ and $u_0$, the solutions converge to 1 behind the front and are globally bounded, improving recent results of Hamel and Ryzhik (arxiv:arXiv:1307.3001). Second, we show that the hydrodynamic limit of the BBM with decay of mass is the solution of the non-local Fisher-KPP equation. We then harness this to obtain several new results concerning the behaviour of the particle system., Comment: 69 pages

We study local optima of the Hamiltonian of the Sherrington-Kirkpatrick model. We compute the exponent of the expected number of local optima and determine the "typical" value of the Hamiltonian., Comment: 20 pages

This work proves new probability bounds relating to the height, width, and size of Galton-Watson trees. For example, if $T$ is any Galton-Watson tree, and $H$, $W$, and $|T|$ are the height, width, and size of $T$, respectively, then $H/W$ has sub-exponential tails and $H/|T|^{1/2}$ has sub-Gaussian tails. Although our methods apply without any assumptions on the offspring distribution, when information is provided about the distribution the method can be adapted accordingly, and always seems to yield tight bounds., Comment: 19 pages

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 Erd\H{o}s-R\'enyi 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 {\L}uczak's result [31] for the Erd\H{o}s-R\'enyi 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 L\'{e}vy 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 Erd\H{o}s-R\'enyi 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., Comment: 31 pages; to appear in Random Structures & Algorithms

We provide new upper bounds for mixing times of general finite Markov chains. We use these bounds to show that the total variation mixing time is robust under rough isometry for bounded degree graphs that are roughly isometric to trees., Comment: 28 pages. Title changed, other minor changes. This version accepted by the Journal of Statistical Physics

We consider fragmentations of an R-tree $T$ driven by cuts arriving according to a Poisson process on $T \times [0, \infty)$, where the first co-ordinate specifies the location of the cut and the second the time at which it occurs. The genealogy of such a fragmentation is encoded by the so-called cut-tree, which was introduced by Bertoin and Miermont for a fragmentation of the Brownian continuum random tree. The cut-tree was generalised by Dieuleveut to a fragmentation of the $\alpha$-stable trees, $\alpha \in (1, 2)$, and by Broutin and Wang to the inhomogeneous continuum random trees of Aldous and Pitman. Remarkably, in all of these cases, the law of the cut-tree is the same as that of the original R-tree. In this paper, we develop a clean general framework for the study of cut-trees of R-trees. We then focus particularly on the problem of reconstruction: how to recover the original R-tree from its cut-tree. This has been studied in the setting of the Brownian CRT by Broutin and Wang, where they prove that it is possible to reconstruct the original tree in distribution. We describe an enrichment of the cut-tree transformation, which endows the cut tree with information we call a consistent collection of routings. We show this procedure is well-defined under minimal conditions on the R-trees. We then show that, for the case of the Brownian CRT and the $\alpha$-stable trees with $\alpha \in (1, 2)$, the original tree and the Poisson process of cuts thereon can both be almost surely reconstructed from the enriched cut-trees. For the latter results, our methods make essential use of the self-similarity and re-rooting invariance of these trees., Comment: 27 pages, 6 figures

For $n\ge 1$, let $T_n$ be a random recursive tree on the vertex set $[n]=\{1,\ldots,n\}$. Let $\mathrm{deg}_{T_n}(v)$ be the degree of vertex $v$ in $T_n$, that is, the number of children of $v$ in $T_n$. Devroye and Lu showed that the maximum degree $\Delta_n$ of $T_n$ satisfies $\Delta_n/\lfloor \log_2 n\rfloor \to 1$ almost surely; Goh and Schmutz showed distributional convergence of $\Delta_n - \lfloor \log_2 n \rfloor$ along suitable subsequences. In this work we show how a version of Kingman's coalescent can be used to access much finer properties of the degree distribution in $T_n$. For any $i\in \mathbb{Z}$, let $X_i^{(n)}=|\{v\in [n]: \mathrm{deg}_{T_n}(v)= \lfloor \log n\rfloor +i\}|$. Also, let $\mathcal{P}$ be a Poisson point process on $\mathbb{R}$ with rate function $\lambda(x)=2^{-x}\cdot \ln 2$. We show that, up to lattice effects, the vectors $(X_i^{(n)},\, i\in \mathbb{Z})$ converge weakly in distribution to $(\mathcal{P}[i,i+1),\, i\in \mathbb{Z})$. We also prove asymptotic normality of $X_i^{(n)}$ when $i=i(n) \to -\infty$ slowly, and obtain precise asymptotics for $\mathbb{P}(\Delta_n - \log_2 n > i)$ when $ i(n) \to \infty$ and $i(n)/\log n$ is not too large. Our results recover and extends the previous results on maximal and near-maximal degrees in random recursive trees., Comment: 15 pages, 3 figures. Revised proof of Proposition 4.5, results unchanged

Let $P$ be the transition matrix of a finite, irreducible and reversible Markov chain. We say the continuous time Markov chain $X$ has transition matrix $P$ and speed $\lambda$ if it jumps at rate $\lambda$ according to the matrix $P$. Fix $\lambda_X,\lambda_Y,\lambda_Z\geq 0$, then let $X,Y$ and $Z$ be independent Markov chains with transition matrix $P$ and speeds $\lambda_X,\lambda_Y$ and $\lambda_Z$ respectively, all started from the stationary distribution. What is the chance that $X$ and $Y$ meet before either of them collides with $Z$? For each choice of $\lambda_X,\lambda_Y$ and $\lambda_Z$ with $\max(\lambda_X,\lambda_Y)>0$, we prove a lower bound for this probability which is uniform over all transitive, irreducible and reversible chains. In the case that $\lambda_X=\lambda_Y=1$ and $\lambda_Z=0$ we prove a strengthening of our main theorem using a martingale argument. We provide an example showing the transitivity assumption cannot be removed for general $\lambda_X,\lambda_Y$ and $\lambda_Z$., Comment: 20 pages

In this paper we explore maximal deviations of large random structures from their typical behavior. We introduce a model for a high-dimensional random graph process and ask analogous questions to those of Vapnik and Chervonenkis for deviations of averages: how "rich" does the process have to be so that one sees atypical behavior. In particular, we study a natural process of Erd\H{o}s-R\'enyi random graphs indexed by unit vectors in $\mathbb{R}^d$. We investigate the deviations of the process with respect to three fundamental properties: clique number, chromatic number, and connectivity. In all cases we establish upper and lower bounds for the minimal dimension $d$ that guarantees the existence of "exceptional directions" in which the random graph behaves atypically with respect to the property. For each of the three properties, four theorems are established, to describe upper and lower bounds for the threshold dimension in the subcritical and supercritical regimes.

Let $D(n,r)$ be a random $r$-out regular directed multigraph on the set of vertices $\{1,\ldots,n\}$. In this work, we establish that for every $r \ge 2$, there exists $\eta_r>0$ such that $\text{diam}(D(n,r))=(1+\eta_r+o(1))\log_r{n}$. Our techniques also allow us to bound some extremal quantities related to the stationary distribution of a simple random walk on $D(n,r)$. In particular, we determine the asymptotic behaviour of $\pi_{\max}$ and $\pi_{\min}$, the maximum and the minimum values of the stationary distribution. We show that with high probability $\pi_{\max} = n^{-1+o(1)}$ and $\pi_{\min}=n^{-(1+\eta_r)+o(1)}$. Our proof shows that the vertices with $\pi(v)$ near to $\pi_{\min}$ lie at the top of "narrow, slippery towers", such vertices are also responsible for increasing the diameter from $(1+o(1))\log_r n$ to $(1+\eta_r+o(1))\log_r{n}$., Comment: 31 pages

We present a probabilistic approach to the core-size in random maps, which yields straightforward and singularity analysis-free proofs of some results of Banderier, Flajolet, Schaeffer and Soria. The proof also yields convergence in distribution of the rescaled size of the k'th largest 2-connected block in a large random map, for any fixed k > 1, to a Fr\'echet-type extreme order statistic. This seems to be a new result even when k=2., Comment: 10 pages, 3 figures. To appear in ALEA - Latin American Journal of Probability and Mathematical Statistics

We show that a uniform quadrangulation, its largest 2-connected block, and its largest simple block jointly converge to the same Brownian map in distribution for the Gromov-Hausdorff-Prokhorov topology. We start by deriving a local limit theorem for the asymptotics of maximal block sizes, extending the result in \cite{BFSS}. The resulting diameter bounds for pendant submaps of random quadrangulations straightforwardly lead to Gromov-Hausdorff convergence. To extend the convergence to the Gromov-Hausdorff-Prokhorov topology, we show that exchangeable "uniformly asymptotically negligible" attachments of mass simply yield, in the limit, a deterministic scaling of the mass measure., Comment: 30 pages, 5 figures. Minor changes for v2

We augment standard branching Brownian motion by adding a competitive interaction between nearby particles. Informally, when particles are in competition, the local resources are insufficient to cover the energetic cost of motion, so the particles' masses decay. In standard BBM, we may define the front displacement at time $t$ as the greatest distance of a particle from the origin. For the model with masses, it makes sense to instead define the front displacement as the distance at which the local mass density drops from $\Theta(1)$ to $o(1)$. We show that one can find arbitrarily large times $t$ for which this occurs at a distance $\Theta(t^{1/3})$ behind the front displacement for standard BBM., Comment: 31 pages

In these expository notes, we describe some features of the multiplicative coalescent and its connection with random graphs and minimum spanning trees. We use Pitman's proof of Cayley's formula, which proceeds via a calculation of the partition function of the additive coalescent, as motivation and as a launchpad. We define a random variable which may reasonably be called the empirical partition function of the multiplicative coalescent, and show that its typical value is exponentially smaller than its expected value. Our arguments lead us to an analysis of the susceptibility of the Erd\H{o}s-R\'enyi random graph process, and thence to a novel proof of Frieze's \zeta(3)-limit theorem for the weight of a random minimum spanning tree., Comment: 42 pages, 4 figures, 25 exercises, 6 open problems

A recent preprint (arXiv:1402.2632) introduced a growth procedure for planar maps, whose almost sure limit is "the uniform infinite 3-connected planar map". A classical construction of Brooks, Smith, Stone and Tutte (1940) associates a squaring of a rectangle (i.e. a tiling of a rectangle by squares) to any to finite, edge-rooted planar map with non-separating root edge. We use this construction together with the map growth procedure to define a growing sequence of squarings of rectangles. We prove the sequence of squarings converges to an almost sure limit: a random infinite squaring of a finite rectangle. This provides a canonical planar embedding of the uniform infinite 3-connected planar map. We also show that the limiting random squaring almost surely has a unique point of accumulation., Comment: 18 pages, 8 figures

We use a growth procedure for binary trees due to Luczak and Winkler, a bijection between binary trees and irreducible quadrangulations of the hexagon due to Fusy, Poulalhon and Schaeffer, and the classical angular mapping between quadrangulations and maps, to define a growth procedure for maps. The growth procedure is local, in that every map is obtained from its predecessor by an operation that only modifies vertices lying on a common face with some fixed vertex. As n tends to infinity, the probability that the n'th map in the sequence is 3-connected tends to 2^8/3^6. The sequence of maps has an almost sure limit G, and we show that G is the distributional local limit of large, uniformly random 3-connected graphs., Comment: 13 pages, 8 figures

Starting from an n-by-n matrix of zeros, choose uniformly random zero entries and change them to ones, one-at-a-time, until the matrix becomes invertible. We show that with probability tending to one as n tends to infinity, this occurs at the very moment the last zero row or zero column disappears. We prove a related result for random symmetric Bernoulli matrices, and give quantitative bounds for some related problems. These results extend earlier work by Costello and Vu [arXiv:math/0606414]., Comment: 31 pages, 1 figure

Assign i.i.d. standard exponential edge weights to the edges of the complete graph K_n, and let M_n be the resulting minimum spanning tree. We show that M_n converges in the local weak sense (also called Aldous-Steele or Benjamini-Schramm convergence), to a random infinite tree M. The tree M may be viewed as the component containing the root in the wired minimum spanning forest of the Poisson-weighted infinite tree (PWIT). We describe a Markov process construction of M starting from the invasion percolation cluster on the PWIT. We then show that M has cubic volume growth, up to lower order fluctuations for which we provide explicit bounds. Our volume growth estimates confirm recent predictions from the physics literature, and contrast with the behaviour of invasion percolation on the PWIT and on regular trees, which exhibit quadratic volume growth., Comment: 39 pages, 1 figure

Consider the minimum spanning tree (MST) of the complete graph with n vertices, when edges are assigned independent random weights. Endow this tree with the graph distance renormalized by n^{1/3} and with the uniform measure on its vertices. We show that the resulting space converges in distribution, as n tends to infinity, to a random measured metric space in the Gromov-Hausdorff-Prokhorov topology. We additionally show that the limit is a random binary R-tree and has Minkowski dimension 3 almost surely. In particular, its law is mutually singular with that of the Brownian continuum random tree or any rescaled version thereof. Our approach relies on a coupling between the MST problem and the Erd\"os-R\'enyi random graph. We exploit the explicit description of the scaling limit of the Erd\"os-R\'enyi random graph in the so-called critical window, established by the first three authors in an earlier paper, and provide a similar description of the scaling limit for a "critical minimum spanning forest" contained within the MST., Comment: 60 pages, 4 figures

"Small worlds" are large systems in which any given node has only a few connections to other points, but possessing the property that all pairs of points are connected by a short path, typically logarithmic in the number of nodes. The use of random walks for sampling a uniform element from a large state space is by now a classical technique; to prove that such a technique works for a given network, a bound on the mixing time is required. However, little detailed information is known about the behaviour of random walks on small-world networks, though many predictions can be found in the physics literature. The principal contribution of this paper is to show that for a famous small-world random graph model known as the Newman--Watts small world, the mixing time is of order (log n)^2. This confirms a prediction of Richard Durrett, who proved a lower bound of order (log n)^2 and an upper bound of order (log n)^3.

We provide simplified proofs for the asymptotic distribution of the number of cuts required to cut down a Galton-Watson tree with critical, finite-variance offspring distribution, conditioned to have total progeny $n$. Our proof is based on a coupling which yields a precise, nonasymptotic distributional result for the case of uniformly random rooted labeled trees (or, equivalently, Poisson Galton-Watson trees conditioned on their size). Our approach also provides a new, random reversible transformation between Brownian excursion and Brownian bridge., Comment: Published in at http://dx.doi.org/10.1214/13-AAP978 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Fix a sequence c=(c_1,...,c_n) of non-negative integers with sum n-1. We say a rooted tree T has child sequence c if it is possible to order the nodes of T as v_1,...,v_n so that for each 1 <= i <= n, v_i has exactly c_i children. Let T be a plane tree drawn uniformly at random from among all plane trees with child sequence c. In this note we prove sub-Gaussian tail bounds on the height (greatest depth of any node) and width (greatest number of nodes at any single depth) of T. These bounds are optimal up to the constant in the exponent when c satisfies c_1^2+...+c_n^2=O(n); the latter can be viewed as a "finite variance" condition for the child sequence., Comment: 8 pages

We shall present an algorithm for determining whether or not a given planar graph H can ever be a subgraph of a 4-regular planar graph. The algorithm has running time O(|H|^{2.5}) and can be used to find an explicit 4-regular planar graph G containing H if such a graph exists. It shall not matter whether we specify that H and G must be simple graphs or allow them to be multigraphs., Comment: 21 pages, 19 figures. To be published in Algorithmica

For a fixed d-regular graph H, a random n-lift is obtained by replacing each vertex v of H by a "fibre" containing n vertices, then placing a uniformly random matching between fibres corresponding to adjacent vertices of H. We show that with extremely high probability, all eigenvalues of the lift that are not eigenvalues of H, have order O(sqrt(d)). In particular, if H is Ramanujan then its n-lift is with high probability nearly Ramanujan. We also show that any exceptionally large eigenvalues of the n-lift that are not eigenvalues of H, are overwhelmingly likely to have been caused by a dense subgraph of size O(|E(H)|)., Comment: 35 pages; substantially revised, constants in the main results improved

We determine, to within O(1), the expected minimal position at level n in certain branching random walks. The walks under consideration have displacement vector (v_1,v_2,...), where each v_j is the sum of j independent Exponential(1) random variables and the different v_i need not be independent. In particular, our analysis applies to the Poisson-Dirichlet branching random walk and to the Poisson-weighted infinite tree. As a corollary, we also determine the expected height of a random recursive tree to within O(1)., Comment: Published in at http://dx.doi.org/10.1214/12-AAP840 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We study the height and width of a Galton--Watson tree with offspring distribution B satisfying E(B)=1, 0 < Var(B) < infinity, conditioned on having exactly n nodes. Under this conditioning, we derive sub-Gaussian tail bounds for both the width (largest number of nodes in any level) and height (greatest level containing a node); the bounds are optimal up to constant factors in the exponent. Under the same conditioning, we also derive essentially optimal upper tail bounds for the number of nodes at level k, for 1 <= k <= n., Comment: 15 pages