Prob-L@b Seminars: Spring 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
9/1/12: Malwina Luczak
(Sheffield) The
supermarket model with arrival rate tending
to 1
There are \(n\)
queues, each with a single
server. Customers arrive in a Poisson
process at rate \(\lambda n\), where \(0 <
\lambda = \lambda (n) < 1\). Upon arrival
each customer selects \(d = d(n) \ge 1\)
servers uniformly at random, and joins the
queue at a least-loaded server among those
chosen. Service times are independent
exponentially distributed random variables
with mean 1.
We will review the literature, including
results of Luczak and McDiarmid (2006), for
the case where \(\lambda\) and \(d\) are
constants independent of \(n\).
We will then investigate the speed of
convergence to equilibrium and the maximum
length of a queue in the equilibrium
distribution when \(\lambda (n) \to 1\) and
\(d(n) \to \infty\) as \(n \to \infty\). This
is joint work with Graham Brightwell.
30/1/12: Mathew Penrose
Connectivity of
\(G(n,r,p)\)
Consider a graph on \(n\)
vertices placed uniformly independently at
random in the unit square, in which any two
vertices distant at most \(r\) apart are
connected by an edge with probability \(p\).
This generalises both the classical random
graph and the random geometric graph. We
discuss the chances of its being
disconnected without having any isolated
vertices, when \(n\) is large, for various
choices of the other parameters.
6/2/12: Martin Barlow (UBC)
The
uniform spanning tree in two dimensions
This talk will discuss
properties of the UST on the Euclidean
lattice, and in particular with the
relation between distance in the tree and
Euclidean distance. These results can then
be applied to study SRW on the UST.
(Martin Barlow and Robert Masson) Note start time: 12.00
13/2/12: Antal Jarai
Minimal configurations
and sandpile measures
We give a brief introduction to the Abelian
sandpile model, and its relationship to
uniform spanning trees. We give a new
simple construction of the sandpile measure
on an infinite graph \(G\), under the sole
assumption that each tree in the Wired
Uniform Spanning Forest on \(G\) has one
end almost surely. For so called
generalized minimal configurations the
limiting probability on \(G\) exists even
without this assumption. We also give
determinantal formulas for minimal
configurations on general graphs in terms
of the transfer current matrix. (Joint work
with N. Werning)
20/2/12: Emmanuel Jacob (Paris
VI)
Second order reflections of the integrated
Brownian motion
27/1/12: Nathalie Eisenbaum
(Paris VI)
Characterization of positively correlated
squared Gaussian processes
When does a centered Gaussian vector
have the property of positive correlation
(also called association or positive
association) ? The answer is found by
Loren Pitt in 1982. In 1991 Steve Evans
raises the problem of the characterization
of the centered Gaussian vectors \((\eta_1,
\eta_2,..., \eta_d)\) such that \((\eta_1^2,
\eta_2^2,..., \eta_d^2)\) is positively
correlated. This talk will present a
solution to that problem.
5/3/12: Stephan Luckhaus
(Leipzig)
Gibbs Gradient Young measures, a stochastic
version of twoscale convergence in
nonlinear elasticity
12/3/12: Vitali Wachtel (LMU,
Munich)
Random walks in cones
We study the asymptotic behaviour of a
multidimensional random walk in a general
cone. We find the tail asymptotics for the
exit time and prove integral and local
limit theorems for a random walk
conditioned to stay in a cone.
19/3/12: Christian
Mönch
Sublogarithmic distances in preferential
attachment networks
Preferential attachment networks with power
law degree sequence undergo a phase
transition when the power law exponent
\(\tau\) changes. For \(\tau>3\) typical
distances in the network are logarithmic in
the size of the network and for \(\tau < 3\)
they are doubly logarithmic. I will discuss
the latter case for a sublinear
preferential attachment model with Poisson
outdegrees and then turn to the critical
behaviour at \(\tau=3\).
26/3/12: Max von Renesse (LMU,
Munich) Hamiltonian Mechanics on
Wasserstein Space and Quantum Fluid Models
Otto's Riemannian
Framework for the Wasserstein space of
probablity measures allows not only for
first order gradient flows but also for
second order Hamiltonian ODEs. As a result
we give a concise representation of the
Schrödinger equation for wave functions as
an instance of Newton's classical law of
motion on Wasserstein space, the two
representations being related by a natural
sympelctic morphism. Introducting friction
leads to dissipative quantum fluid models
such as the Quantum Navier Stokes equation,
which was derived as a model for a tagged
particle in a many body quantum system.
Partially based on joint works with
A. Jüngel and P. Fuchs (Vienna)
16/4/12: Richard Cowan
(Sydney) A
topological identity for the convex hulls
of \(n\) arbitrary points in
\(\mathbb{R}^d\) - and its role in the
study of convex hulls of random points
The author will speak
about a new topologically identity that he
discovered in 2007. It concerns convex
hulls of \(n\) arbitrarily positioned points in
\(d\)-dimensional Euclidean space. The identity
gives insights into the more specialised
situation where the points are randomly
placed.
Whilst the talk is being presented to the
Probability Group, it should be of interest
to others in the department.
21/5/12: Rafał
Łochowski (Warsaw School of Economics)
Truncated variation of a stochastic process
- its optimality for processes with cadlag
trajectories and its characterisations for
semimartingales, diffusions and Wiener
process.
For a given
function \(f : [a; b] \to R\), we define its
truncated variation at the level \(c >0\),
\(TV^c (f , [a; b])\), as the smallest
possible and attainable total variation of
a function uniformly approximating \(f\)
with accuracy \(c/2\). It appears, that for
\(f\) being a cadlag function its truncated
variation is always finite, in opposite to
total variation, which is a limit value of
\(TV^c (f , [a; b])\) as \(c\) tends to
\(0\) and may be infinite.
Together
with \(TV^c (f , [a; b])\), we define two other
functional related - upward and downward
truncated variations. This may be viewed as
a generalization of Hahn-Jordan
decomposition of a function with finite
total variation.
In the second part of my talk I will
present briefly results on behaviour of
\(TV^c (X, [0; T])\) for semimartingales,
diffusions and Wiener process. For \(X\) being
a semimartingale, I will present results on
a.s. functional convergence of the process
\(c TV^c (X, [0; t])\) on the interval \(0 \leq t
\leq T\) as \(c\) tends to \(0\); for \(X\) being a
diffusion, I will present results on weak
functional convergence of the process \(TV^c
(X, [0; t]) - _t/c\) on the interval \(0
\leq t \leq T\) as c tends to 0 and for
Wiener process with drift I will present
full characterisation of \(TV^c (X, [0; T])\)
via its Laplace transform. The majority of
my talk will be based on the join work with
Piotr Milos.
28/5/12: Christoph
Höggerl
Model-independent no-arbitrage conditions
for American Put options
Suppose European Put options with fixed
maturity and for a finite number of strikes
are traded in the market and these prices
are consistent with no (model-independent)
arbitrage We investigate necessary and
sufficient conditions on the American Put
option prices corresponding to the absence
of arbitrage in the extended market, where
not only European, but also American
options can be traded.
13/6/12: Stavros Vakeroudis
(Manchester)
Windings of planar processes
We first study the distribution of
several first hitting times for the
continuous winding process associated with
the planar Brownian motion. To obtain
analytical results, we use and Bougerol's
celebrated identity in law. This allows us
to characterize the laws of the hitting
times of the bounadries of a cone for
planar Brownian motion. We give some
applications, namely: a new
non-computational proof of Spitzer's
asymptotic theorem, some integrability
properties for these hitting times, similar
results for the (planar) complex-valued
Ornstein-Uhlenbeck processes case.
Finally, we shall discuss the windings of
planar stable processes, where the approach
is different than in the Brownian motion
case.
Rearranged: Note new date and time:
Wednesday 10.15
2/7/12: Peter Windridge
The
SIR process on random graphs with given
degree sequences
The SIR process is a model for disease
spreading through a network. The network
topology is given by a graph G. Vertices
are either susceptible, infective or
recovered. Infectives transmit the disease
to each susceptible neighbour at rate b > 0
and recover at rate r > 0. Usually one
takes the graph size to infinity and
studies e.g. the probability that the
disease dies out.
Taking G itself to be random is both
mathematically interesting and relevant to
real world networks. Recent results have
focused on graphs chosen uniformly subject
to having prescribed vertex degrees. We'll
review these results and discuss a possible
way to extend them.
CANCELLED
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