# Trinity Term 2016

### TRINITY TERM 2016

Probability Workshops are held on Mondays from 12:00 - 1:15 pm during term time (unless indicated otherwise) in L4 at The Mathematical Institute, Andrew Wiles Building, Radcliffe Observatory Quarter.

#### Week 1 - Thursday 25th April

Speaker:   Christina Goldschmidt, Department of Statistics, University of Oxford
Title:          The cut-tree transform and its inverse
Abstract:  A natural, and much-studied, fragmentation operation on (random) discrete trees is as follows: pick an edge uniformly at random and remove it; repeat until the tree is reduced to a collection of isolated vertices. The genealogy of this fragmentation is encoded by the so-called cut-tree. Here, the vertices of the tree represent blocks which appear at some point in the fragmentation process, with one block being the child of another one if it is obtained by a splitting event.

In this talk, I will focus on the continuum analogue of the cut-tree transform, where we think now about fragmenting a continuum random tree.  In the case of the Brownian continuum random tree, split apart by to a Poisson process of cuts along its length, this fragmentation is one of the foundational examples of a self-similar fragmentation, for which a suitable time-reversal is the additive coalescent.  In that setting, a remarkable result due to Bertoin & Miermont is that the cut-tree is again a Brownian continuum random tree.  This equality in distribution has been shown to extend to a few other particular cases: the stable trees fragmented at nodes (Dieuleveut) and certain inhomogeneous continuum random trees (Broutin & Wang).

In my talk, I will set up the cut-tree transform in some generality and then address the problem of reconstruction: is it possible to recover the original tree from its cut-tree (and what information do we need to carry in order to make this possible)?  We show that the Brownian CRT and the other stable trees can be reconstructed almost surely. In particular, this yields new examples of operations on R-trees for which the law of a stable tree is a fixed point.

This is joint work with Louigi Addario-Berry and Daphne Dieuleveut.

#### Week 2 - Monday 2nd May

Bank Holiday - No seminar

#### Week 3 - Monday 9th May

Speaker:    Owen Jones, The University of Melbourne
Title:         Coalescing runoff flows
Abstract:   The volume of catchment discharge that reaches a stream via the overland flow path is critical for water quality prediction, because it is via this pathway that most particulate pollutants are generated and transported to the stream channel, via surface erosion processes. When it rains, spatial variation in the soil infiltration rate leads to the formation and reabsorption of rivulets on the surface, and local topography determines the coalescence of rivulets.

We consider the question of how coalescence facilitates overland flow in two ways. Firstly we take a highly abstracted version of the problem, in which the drainage pattern is represented by a Galton-Watson tree. We show that as the rate of rainfall increases there is a distinct phase-change: when there is no stream coalescence the critical point occurs when the rainfall rate equals the infiltration rate, but when we allow coalescence the critical point occurs when the rainfall rate is less than the infiltration rate, and increasing the amount of coalescence increases the total expected runoff.

Secondly we fit a model for coalescing runoff to field data collected over a period of months from a burnt hillslope in the Victorian alps in SE Australia. The hillslope is discretised using a hexagonal lattice, and local topography is modelled by randomly perturbing the drainage pattern (which is derived from satellite elevation data). More perturbation corresponds to a rougher surface and more coalescence. The many latent variables mean the model has an intractable likelihood, however it is easily simulated, allowing us to fit the model using Approximate Bayesian Computation (ABC).

#### Week 4 - Monday 16th May

Speaker:   Sanchayan Sen, Department of Statistics, University of Oxford
Title:        Random graphs: scaling limits and universality
Abstract:  One major conjecture in probabilistic combinatorics, formulated by statistical physicists using non-rigorous arguments and enormous simulations in the early 2000s, is as follows:  for a wide array of random graph models  on    vertices and degree exponent  , typical distance 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  . In other words, the degree exponent determines the universality class the random graph belongs to. There is also mounting evidence to believe that for a wide class of random graph models, components at criticality and various other structures on the graphs viewed as metric measure spaces converge, after scaling the graph distance, to some random fractals, and the degree exponent   determines the universality class of the limiting object. We report on recent progress in proving these conjectures.

#### Week 5 - Monday 23rd May

Speaker:   Franz Rembart, Department of Statistics, University of Oxford
Title:           Recursive construction of continuum random trees and a binary embedding of stable trees
Abstract:   We introduce a general recursive method to construct continuum random trees (CRTs) from i.i.d. copies of a string of beads, that is, any random interval equipped with a random discrete measure. We prove the existence of these CRTs as a new application of the fixpoint method formalised in high generality by Aldous and Bandyopadhyay. We apply this recursive method to "embed" Duquesne and Le Gall's stable tree into a binary compact CRT in a way that solves an open problem posed by Goldschmidt and Haas. Some of these developments are carried out in a space of $\infty$-marked metric spaces generalising Miermont's notion of a $k$-marked metric space.

#### Week 6 - Monday 30th May

Bank Holiday - No Seminar

No Seminar

#### Week 8 - Monday 13th June

Speaker:    Daniel Diaz, University of Miami
Title:           Some properties of the stable marriage of Poisson and Lebesgue
Abstract:   The stable marriage of Poisson and Lebesgue was first developed as a tool to produce a shift coupling of a Poisson process such that the coupling has the same law as the Palm version of the original Poisson process. Since its inception it has produced several interesting questions. The model depends on two parameters, the intensity of the Poisson process and another one called "appetite". In this talk we focus on some of the properties of this model, with a particular emphasis on percolation and tail bounds, both with constant and random appetites.

Previous Workshops: HT16; MT15; TT15HT15; MT14; TT14;  HT14;   MT13