Algebraic trees versus metric trees as states of stochastic processes |
Prof. Anita Winter |
Donnerstag, 2017-10-19 17:00Uhr |
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In this talk we are interested in limit objects of graph-theoretic trees as the number of vertices goes to infinity. Depending on which notion of convergence we choose different objects are obtained. One notion of convergence with several applications in different areas is based on encoding trees as metric measure spaces and then using the Gromov-weak topology. Apparently this notion is problematic in the construction of scaling limits of tree-valued Markov chains whenever the metric and the measure have a different scaling regime. We therefore introduce the notion of algebraic measure trees which capture only the tree structure but not the metric distances. Convergence of algebraic measure trees will then rely on weak convergence of the random shape of a subtree spanned a sample of finite size. We will be particularly interested in binary algebraic measure trees which can be encoded by triangulations of the circle. We will show that in the subspace of binary algebraic measure trees sample shape convergence is equivalent to Gromov-weak convergence when we equip the algebraic measure tree with an intrinsic metric coming from the branch point distribution. We will illustrate this with the example of a Markov chain arising in phylogeny whose mixing behavior was studied in detail by Aldous (2000) and Schweinsberg (2001). (based on joint work with Wolfgang Löhr and Leonid Mytnik) |