The local weak limit of the minimum spanning tree of the complete graph

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 figur

Most trees are short and fat

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 page

Growing random 3-connected maps, or comment s'enfuir de l'hexagone

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 figure

The spectrum of random lifts

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 improve

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

Random infinite squarings of rectangles

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 figure

The front location in BBM with decay of mass

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 page

Total progeny in killed branching random walk

We consider a branching random walk for which the maximum position of a particle in the n'th generation, M_n, has zero speed on the linear scale: M_n/n --> 0 as n --> infinity. We further remove ("kill") any particle whose displacement is negative, together with its entire descendence. The size $Z$ of the set of un-killed particles is almost surely finite. In this paper, we confirm a conjecture of Aldous that Exp[Z] < infinity while Exp[Z log Z]=infinity. The proofs rely on precise large deviations estimates and ballot theorem-style results for the sample paths of random walks.Comment: 21 pages, 1 figur

High degrees of random recursive trees

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 unchange

The scaling limit of random simple triangulations and random simple quadrangulations

Let $M_n$ be a simple triangulation of the sphere $S^2$, drawn uniformly at random from all such triangulations with n vertices. Endow $M_n$ with the uniform probability measure on its vertices. After rescaling graph distance on $V(M_n)$ by $(3/(4n))^{1/4}$, the resulting random measured metric space converges in distribution, in the Gromov-Hausdorff-Prokhorov sense, to the Brownian map. In proving the preceding fact, we introduce a labelling function for the vertices of $M_n$. Under this labelling, distances to a distinguished point are essentially given by vertex labels, with an error given by the winding number of an associated closed loop in the map. We establish similar results for simple quadrangulations.Comment: 47 pages, 10 figures Revised argument in section 6, section 4 rewritte