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Random trees and tree-valued stochastic processes are of particular importance in combinatorics, computer science, phylogenetics, and mathematical population genetics. Using the framework of abstract "tree-like" metric spaces (so-called real trees) and ideas from metric geometry such as the Gromov-Hausdorff distance, Evans and his collaborators have recently pioneered an approach to studying the asymptotic behaviour of such objects when the number of vertices goes to infinity. These notes survey the relevant mathematical background and present some selected applications of the theory.
Trees (Graph theory) --- Markov processes --- Stochastic processes --- Metric spaces --- Hausdorff measures --- Dirichlet forms --- Phylogeny --- Evolutionary genetics --- Mathematical Statistics --- Algebra --- Mathematics --- Physical Sciences & Mathematics --- Mathematical models --- Phylogenèse --- Génétique évolutive --- Processus stochastiques --- Modèles mathématiques --- Genetic evolution --- Animal phylogeny --- Animals --- Phylogenetics --- Phylogeny (Zoology) --- Forms, Dirichlet --- Spaces, Metric --- Mathematics. --- Geometry. --- Probabilities. --- Combinatorics. --- Probability Theory and Stochastic Processes. --- Combinatorics --- Mathematical analysis --- Euclid's Elements --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Math --- Science --- Evolution (Biology) --- Genetics --- Biology --- Forms (Mathematics) --- Generalized spaces --- Set theory --- Topology --- Measure theory --- Graph theory --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities
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Discrete mathematics --- Genetics --- Evolutionary genetics. --- Mutation (Biology) --- Genetic recombination. --- Génétique évolutive --- Mutation (Biologie) --- Recombinaison génétique --- 51 <082.1> --- Mathematics--Series --- Génétique évolutive --- Recombinaison génétique --- Evolutionary genetics --- Genetic recombination --- Variation (Biology) --- Recombination, Genetic --- Chromosomes --- Recombinant DNA --- Genetic evolution --- Evolution (Biology)
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Die jüngsten Entwicklungen zeigen, dass sich Wahrscheinlichkeitsverfahren zu einem sehr wirkungsvollen Werkzeug entwickelt haben, und das auf so unterschiedlichen Gebieten wie statistische Physik, dynamische Systeme, Riemann'sche Geometrie, Gruppentheorie, harmonische Analyse, Graphentheorie und Informatik. Recent developments show that probability methods have become a very powerful tool in such different areas as statistical physics, dynamical systems, Riemannian geometry, group theory, harmonic analysis, graph theory and computer science. This volume is an outcome of the special semester 2001 - Random Walks held at the Schrödinger Institute in Vienna, Austria. It contains original research articles with non-trivial new approaches based on applications of random walks and similar processes to Lie groups, geometric flows, physical models on infinite graphs, random number generators, Lyapunov exponents, geometric group theory, spectral theory of graphs and potential theory. Highlights are the first survey of the theory of the stochastic Loewner evolution and its applications to percolation theory (a new rapidly developing and very promising subject at the crossroads of probability, statistical physics and harmonic analysis), surveys on expander graphs, random matrices and quantum chaos, cellular automata and symbolic dynamical systems, and others. The contributors to the volume are the leading experts in the area. The book will provide a valuable source both for active researchers and graduate students in the respective fields.
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