Stochastic processes and geometry of random networks

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My research focuses on the interface between discrete and continuous objects, and on connections between the geometry and topological properties of structures, both random and ordered, and the behaviour of stochastic processes on them.***Planar maps.  A main aim here is the development of new tools for analysis of random planar maps and quantum gravity, a thriving field at the intersection of probability, statistical physics, combinatorics and complex analysis.  A key idea is view planar maps as a surface rather than as a metric space, whether by endowing the maps with a conformal structure, or by using Koebe's circle packing theorem to embed the maps in R2.  This leads to central open problems in this field such as determination of the speed exponent and scaling limit for the simple random walk on random planar maps, and to the KPZ identity, a conjectural formula relating dimensions of certain random sets in random maps and in Z2, where computing the dimensions are notoriously hard problems.***Scaling of maps.  A second aim is to extend recent results giving the scaling limits of random maps to other classes of maps.  While some types of maps have been analyzed, there are still very few tools for studying the large scale structure of more general random planar maps.  In particular, I plan to study maps endowed with some statistical physical model such as spanning trees or independent sets.****Geodesic networks.  A third aim is to better understand the structure of geodesics in the Brownian map.  It is known that in the Brownian map geodesics to a point typically coalesce midway, and that almost no point is in the interior of any geodesic.  I intend to extend this, and study geodesics network, and in particular show that the union of all geodesics without their endpoints has dimension 1.****High genus maps.  A long standing conjecture is that random maps of high genus have a distributional limit, and there is a conjectured distribution for the limit.  In order to prove this I will build on work with Ray and others on unicellular maps.  The next step is to understand the number of graph homeomorphisms from the unicellular maps to Z.****Random walks and geometry of groups.  Amenability and the Liouville property (every bounded harmonic function is constant), are two fundamental geometric properties of some groups.  It is notoriously hard to determine in general whether a group is amenable and if it is Liouville.  A major goal is to prove that being Liouville does not depend on the choice of a generating set of a group.  I proceed by studying random walks on the groups, and on Schreier graphs for actions of the groups.  By showing that the Schreier graphs are recurrent we can prove that certain groups are Liouville, and are thus also amenable.  I will expand the applicability of these methods to other groups for which amenability and Liouville are not known, such as Thompson's group, and the interval exchange group, and certain automaton groups conjectured by Sidki to be amenable.**