Prof. Anita Winter:
Algebraic trees versus metric trees as states of stochastic processes
Time and place
Thursday, 19.10.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
Abstract
In this talk we are interested in limit objects of graph-theoretic trees\nas the number of vertices goes to infinity. Depending on which notion of\nconvergence we choose different objects are obtained.\n\nOne notion of convergence with several applications in different areas is\nbased on encoding trees as metric measure spaces and then using the\nGromov-weak topology. Apparently this notion is problematic in the\nconstruction of scaling limits of tree-valued Markov chains whenever the\nmetric and the measure have a different scaling regime. We therefore\nintroduce the notion of algebraic measure trees which capture only the tree\nstructure but not the metric distances.\nConvergence of algebraic measure trees will then rely on weak convergence\nof the random shape of a subtree spanned a sample of finite size.\nWe will be particularly interested in binary algebraic measure trees which\ncan be encoded by triangulations of the circle. We will show that in the\nsubspace of binary algebraic measure trees sample shape convergence is\nequivalent to Gromov-weak convergence when we equip the algebraic measure\ntree with an intrinsic metric coming from the branch point distribution.\nWe will illustrate this with the example of a Markov chain arising in\nphylogeny whose mixing behavior was studied in detail by Aldous (2000) and\nSchweinsberg (2001).\n\n (based on joint work with Wolfgang Löhr and Leonid Mytnik)\n