The multiplicative coalescent, inhomogeneous continuum random trees, and new universality classes for critical random graphs

One major open conjecture in the area of critical random graphs, formulated by statistical physicists, and supported by a large amount of numerical evidence over the last decade [23, 24, 28, 63] is as follows: for a wide array of random graph models with degree exponent $\tau\in (3,4)$, distances between typical points both within maximal components in the critical regime as well as on the minimal spanning tree on the giant component in the supercritical regime scale like $n^{(\tau-3)/(\tau-1)}$. In this paper we study the metric space structure of maximal components of the multiplicative coalescent, in the regime where the sizes converge to excursions of L\'evy processes "without replacement" [10], yielding a completely new class of limiting random metric spaces. A by-product of the analysis yields the continuum scaling limit of one fundamental class of random graph models with degree exponent $\tau\in (3,4)$ where edges are rescaled by $n^{-(\tau-3)/(\tau-1)}$ yielding the first rigorous proof of the above conjecture. The limits in this case are compact "tree-like" random fractals with finite fractal dimensions and with a dense collection of hubs (infinite degree vertices) a finite number of which are identified with leaves to form shortcuts. In a special case, we show that the Minkowski dimension of the limiting spaces equal $(\tau-2)/(\tau-3)$ a.s., in stark contrast to the Erd\H{o}s-R\'{e}nyi scaling limit whose Minkowski dimension is 2 a.s. It is generally believed that dynamic versions of a number of fundamental random graph models, as one moves from the barely subcritical to the critical regime can be approximated by the multiplicative coalescent. In work in progress, the general theory developed in this paper is used to prove analogous limit results for other random graph models with degree exponent $\tau\in (3,4)$.Comment: 71 pages, 5 figures, To appear in Probability Theory and Related Field

Tree-valued Fleming-Viot dynamics with mutation and selection

The Fleming-Viot measure-valued diffusion is a Markov process describing the evolution of (allelic) types under mutation, selection and random reproduction. We enrich this process by genealogical relations of individuals so that the random type distribution as well as the genealogical distances in the population evolve stochastically. The state space of this tree-valued enrichment of the Fleming-Viot dynamics with mutation and selection (TFVMS) consists of marked ultrametric measure spaces, equipped with the marked Gromov-weak topology and a suitable notion of polynomials as a separating algebra of test functions. The construction and study of the TFVMS is based on a well-posed martingale problem. For existence, we use approximating finite population models, the tree-valued Moran models, while uniqueness follows from duality to a function-valued process. Path properties of the resulting process carry over from the neutral case due to absolute continuity, given by a new Girsanov-type theorem on marked metric measure spaces. To study the long-time behavior of the process, we use a duality based on ideas from Dawson and Greven [On the effects of migration in spatial Fleming-Viot models with selection and mutation (2011c) Unpublished manuscript] and prove ergodicity of the TFVMS if the Fleming-Viot measure-valued diffusion is ergodic. As a further application, we consider the case of two allelic types and additive selection. For small selection strength, we give an expansion of the Laplace transform of genealogical distances in equilibrium, which is a first step in showing that distances are shorter in the selective case.Comment: Published in at http://dx.doi.org/10.1214/11-AAP831 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Random Metric Spaces and Universality

WWe define the notion of a random metric space and prove that with probability one such a space is isometricto the Urysohn universal metric space. The main technique is the study of universal and random distance matrices; we relate the properties of metric (in particulary universal) space to the properties of distance matrices. We show the link between those questions and classification of the Polish spaces with measure (Gromov or metric triples) and with the problem about S_{\infty}-invariant measures in the space of symmetric matrices. One of the new effects -exsitence in Urysohn space so called anarchical uniformly distributed sequences. We give examples of other categories in which the randomness and universality coincide (graph, etc.).Comment: 38 PAGE

Distance covariance in metric spaces

We extend the theory of distance (Brownian) covariance from Euclidean spaces, where it was introduced by Sz\'{e}kely, Rizzo and Bakirov, to general metric spaces. We show that for testing independence, it is necessary and sufficient that the metric space be of strong negative type. In particular, we show that this holds for separable Hilbert spaces, which answers a question of Kosorok. Instead of the manipulations of Fourier transforms used in the original work, we use elementary inequalities for metric spaces and embeddings in Hilbert spaces.Comment: Published in at http://dx.doi.org/10.1214/12-AOP803 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

A mixing tree-valued process arising under neutral evolution with recombination

The genealogy at a single locus of a constant size $N$ population in equilibrium is given by the well-known Kingman's coalescent. When considering multiple loci under recombination, the ancestral recombination graph encodes the genealogies at all loci in one graph. For a continuous genome $G$, we study the tree-valued process $(T^N_u)_{u\in G}$ of genealogies along the genome in the limit $N\to\infty$. Encoding trees as metric measure spaces, we show convergence to a tree-valued process with cadlag paths. In addition, we study mixing properties of the resulting process for loci which are far apart.Comment: 25 pages, 3 figure

On choosing and bounding probability metrics

When studying convergence of measures, an important issue is the choice of probability metric. In this review, we provide a summary and some new results concerning bounds among ten important probability metrics/distances that are used by statisticians and probabilists. We focus on these metrics because they are either well-known, commonly used, or admit practical bounding techniques. We summarize these relationships in a handy reference diagram, and also give examples to show how rates of convergence can depend on the metric chosen.Comment: To appear, International Statistical Review. Related work at http://www.math.hmc.edu/~su/papers.htm