In this note, we provide a new characterization of Aldous' Brownian continuum random tree as the unique fixed point of a certain natural operation on continuum trees (which gives rise to a recursive distributional equation). We also show that this fixed point is attractive.
As mentioned in Subsection , Aldous @cite_24 shows that, in a sense, we can reverse'' the operator @math . Indeed, we can decompose a BCRT by picking three uniform points and splitting at the branch-point between them; we obtain three independent BCRT's, Brownian-rescaled by @math . Each of these subtrees is doubly marked, one mark being the original uniform point and the other being the former branch-point. Perhaps a more natural way of phrasing the reversal, which yields only a single mark in each subtree, would be to pick each of the branch-points in the tree with probability given by 6 times the product of the masses of the subtrees into which removal of that branch-point splits the tree.
@cite_24
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We study supercritical branching Brownian motion on the real line starting at the origin and with constant drift @math . At the point @math , we add an absorbing barrier, i.e. individuals touching the barrier are instantly killed without producing offspring. It is known that there is a critical drift @math , such that this process becomes extinct almost surely if and only if @math . In this case, if @math denotes the number of individuals absorbed at the barrier, we give an asymptotic for @math as @math goes to infinity. If @math and the reproduction is deterministic, this improves upon results of [L. Addario-Berry and N. Broutin (2009), http: arxiv.org abs 0908.1083v1 ] and [E. A " dekon (2009), http: arxiv.org abs 0911.0877v1 ] on a conjecture by David Aldous about the total progeny of a branching random walk. The main technique used in the proofs is analysis of the generating function of @math near its singular point @math , based on classical results on some complex differential equations. Consider a uniform random rooted tree on vertices labelled by @math , with edges directed towards the root. We imagine that each node of the tree has space for a single car to park. A number @math of cars arrive one by one, each at a node chosen independently and uniformly at random. If a car arrives at a space which is already occupied, it follows the unique path oriented towards the root until it encounters an empty space, in which case it parks there; if there is no empty space, it leaves the tree. Consider @math and let @math denote the event that all @math cars find spaces in the tree. Lackner and Panholzer proved (via analytic combinatorics methods) that there is a phase transition in this model. Then if @math , we have @math , whereas if @math we have @math . We give a probabilistic explanation for this phenomenon, and an alternative proof via the objective method. Along the way, we are led to consider the following variant of the problem: take the tree to be the family tree of a Galton-Watson branching process with Poisson(1) offspring distribution, and let an independent Poisson( @math ) number of cars arrive at each vertex. Let @math be the number of cars which visit the root of the tree. Then for @math , we have @math , whereas for @math , we have @math . This discontinuous phase transition turns out to be a generic phenomenon in settings with an arbitrary offspring distribution of mean at least 1 for the tree and arbitrary arrival distribution. Consider a Markov chain on an infinite tree T = (V,E) rooted at . In such a chain, once the initial root state ( ) is chosen, each vertex iteratively chooses its state from the one of its parent by an application of a Markov transition rule (and all such applications are independent). Let µj denote the resulting measure for ( ) = j. µj is defined on configurations = ( (x))x2V 2 A V , where A is some finite set. Let µ n denote the restriction of µ to the sigma-algebra generated by the variables (x) where x is at distance exactly n from . Letting n = maxi,j2A dTV (µ n ,µ n ), where dTV denotes total variation distance, we say that the reconstruction problem is solvable if liminfn!1 n > 0. Reconstruction solvability roughly means that the n’th level of the tree contains a non-vanishing amount of information on the root of the tree, as n ! 1. In this paper we study the problem of robust reconstruction. Let be a non degenerate distribution on A and > 0. Let be chosen according to µ n and 0 be obtained from by letting for each node independently (v) = 0 (v) with probability 1 and 0 (v) to be an independent sample from otherwise. We denote by µ n [, ] the resulting measure on 0 . The measure µ n [, ] is a perturbation of the measure µ n . Letting n(, ) = maxi,j2A dTV (µ n [, ],µ n [, ]), we say that the reconstruction problem is -robust-solvable if liminfn!1 n(, ) > 0 for all 0 1, where 2(M) is the second largest eigenvalue of M (in absolute value), then for all non-degenerate , the reconstruction problem is -robust-solvable. We prove a converse and show that the reconstruction problem is not -robust-solvable if B| 2(M)| 2 < 1. This proves a conjecture by the second author and Peres. We also consider other models of noise and general trees. Statistical phylogenetic inference methods use tree rearrangement operations such as subtree–prune–regraft (SPR) to perform Markov chain Monte Carlo (MCMC) across tree topologies. The structure of the graph induced by tree rearrangement operations is an important determinant of the mixing properties of MCMC, motivating the study of the underlying SPR graph in greater detail. In this paper, we investigate the SPR graph of rooted trees (rSPR graph) in a new way: by calculating the Ricci–Ollivier curvature with respect to uniform and Metropolis–Hastings random walks. This value quantifies the degree to which a pair of random walkers from specified points move towards each other; negative curvature means that they move away from one another on average, while positive curvature means that they move towards each other. In order to calculate this curvature, we develop fast new algorithms for rSPR graph computation. We then develop formulas characterizing how the number of rSPR neighbors of a tree changes after an rSPR operation is applied to that tree. These give bounds on the curvature, as well as a flatness-in-the-limit theorem indicating that paths of small topology changes are easy to traverse. However, we find that large topology changes (i.e. moving a large subtree) give pairs of trees with negative curvature. We show using simulation that mean access time distributions depend on distance, degree, and curvature, demonstrating the relevance of these results to stochastic tree search.