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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