Department of Mathematical Sciences

Prob-L@B

(Probability Laboratory at Bath)

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Alex Cox
Email: A.M.G.Cox@bath.ac.uk

Prob-L@b Seminars: Winter 2012

Our seminars our usually held at 12.15 p.m. on Mondays in room 4W 1.7 . If you wish to find out more, please contact one of the organisers. The speakers are internal (Bath) unless otherwise stated. Details of previous semesters can be found here

8/10/12: Peter Mörters

Clustering in spatial preferential attachment networks

I define a class of growing networks in which new nodes are given a spatial position and are connected to existing nodes with a probability mechanism favouring short distances and high degrees. The competition of preferential attachment and spatial clustering gives this model a range of interesting properties. Empirical degree distributions converge to a limiting power law, and the average clustering coefficient of the networks converges to a positive limit. A phase transition occurs in the global clustering coefficients and empirical distribution of edge lengths. The talk is based on joint work with Emmanuel Jacob (ENS Lyon).

15/10/12: Martin Klimmek (Oxford)

From inverse optimal stopping to BLJ embeddings

Connections between an old earthwork problem (transport material from a distribution at the depot to a target distribution) and the theory of model-independent pricing/hedging have led to new interest in Skorokhod embedding theory and generalized convex analysis. Underlying the burgeoning field of optimal martingale transport is a classical problem in martingale theory/mathematical finance: find extremal martingales consistent with given marginals/call prices at fixed times. The two inverse problems considered here differ in two ways. The time-horizon is either random or infinite, and solutions must be diffusions: We begin by constructing diffusions consistent with given value functions for perpetual horizon stopping problems using generalized convex analysis and basic one-dimensional diffusion theory. We move on to characterize the diffusions with a given marginal at a random time, the solution leading us back to Skorokhod embeddings.

22/10/12: Maren Eckhoff

Surviving near criticality in a preferential attachment network

We study a dynamical network model in which at every time step a new vertex is added and attached to every existing vertex independently with a probability proportional to a concave function of its current degree. Locally, this network can be approximated by a typed branching random walk (BRW) with an absorbing barrier.
We investigate the size of the giant component in the network near criticality, or equivalently, the asymptotics of the survival probability of the BRW under a changing offspring distribution. The talk is based on joint work with Peter Mörters.

29/10/12: Matt Roberts (Warwick)

Intermittency in branching random walk in random environment

Over the last 20 years mathematicians have proved rigorously that the parabolic Andreson model shows the intermittency behaviour predicted by physicists. We shall see that a branching random walk in Pareto random environment displays the same qualitative behaviour, but with several important differences. This is work in progress with Marcel Ortgiese.

5/11/12: Edward Crane (Bristol)

Antichains in random partial orders

19/11/12: Stefan Adams (Warwick)

Random Field of Gradients

Random fields of gradients are a class of model systems arising in the studies of random interfaces, random geometry, field theory, and elasticity theory. These random objects pose challenging problems for probabilists as even an a priori distribution involves strong correlations. Gradient fields are likely to be an universal class of models combining probability, analysis and physics in the study of critical phenomena. They emerge in the following three areas, effective models for random interfaces, Gaussian Free Fields (scaling limits), and mathematical models for the Cauchy-Born rule of materials, i.e., a microscopic approach to nonlinear elasticity. The latter class of models requires that interaction energies are non-convex functions of the gradients. Open problems include unicity of Gibbs measures and strict convexity of the free energy. We present in the talk a first break through for the free energy at low temperatures using Gaussian measures and rigorous renormalisation group techniques. In addition we show that the correlation functions have Gaussian decay properties despite the fact having non-convex interaction potentials. The key ingredient is a finite range decomposition for parameter dependent families of Gaussian measures. If time permits to discuss the connection of Gaussian Free Fields and interlacements.

CANCELLED - NEW DATE TBC

26/11/12: Jesse Goodman (Leiden)

The gaps left by Brownian motion on a torus

Run a Brownian motion on a torus for a long time. How large are the random gaps left behind when the path is removed?
In three (or more) dimensions, we find that there is a deterministic spatial scale common to all the large gaps anywhere in the torus. Moreover, we can identify whether a gap of a given shape is likely to exist on this scale, in terms of a single parameter, the classical (Newtonian) capacity. I will describe why this allows us to identify a well-defined "component" structure in our random porous set.

3/12/12: Bálint Tóth (Bristol)

Diffusivity of random walk in divergence free drift field in 2d

10/12/12: Tony Shardlow

Pathwise approximation of SDEs and adaptive time stepping

We give a pathwise convergence analysis of one step integrators of SDEs. The analysis is motivated by rough path theory and applies to a general class of one steps methods subject to a pathwise bound on the sum of the truncation errors. We show how the method is applied to the Euler method. We take particular interest in showing the pathwise convergence of an adaptive time stepping method based on bounded diffusions and establish that the constant in the pathwise error can be controlled.

14/1/13: Stefan Adams (Warwick)

Random Field of Gradients

Random fields of gradients are a class of model systems arising in the studies of random interfaces, random geometry, field theory, and elasticity theory. These random objects pose challenging problems for probabilists as even an a priori distribution involves strong correlations. Gradient fields are likely to be an universal class of models combining probability, analysis and physics in the study of critical phenomena. They emerge in the following three areas, effective models for random interfaces, Gaussian Free Fields (scaling limits), and mathematical models for the Cauchy-Born rule of materials, i.e., a microscopic approach to nonlinear elasticity. The latter class of models requires that interaction energies are non-convex functions of the gradients. Open problems include unicity of Gibbs measures and strict convexity of the free energy. We present in the talk a first break through for the free energy at low temperatures using Gaussian measures and rigorous renormalisation group techniques. In addition we show that the correlation functions have Gaussian decay properties despite the fact having non-convex interaction potentials. The key ingredient is a finite range decomposition for parameter dependent families of Gaussian measures. If time permits to discuss the connection of Gaussian Free Fields and interlacements.

** Note unusual time: 13:15 **

 

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