Statistical Laboratory
Probability Seminars
Easter Term 2002
All interested are encouraged
to take part to the full by presenting their ideas and discussing
those of others.
Talks will be in Meeting Room 12 of the
Centre for Mathematical Sciences, Pavilion D.
Tuesday 30 April
2pm Yuri Suhov
Quantum entropy and quantum coding
Abstract. In classical information coding, we want
to encode `messages' $x_1^n$ $=$ $(x_1,\ldots ,x_n)$,
of length $n\to\infty$,
generated by a `source' represented by an ergodic
process $X_{-\infty}^\infty$ with values from a
`source alphabet' $V$ and probabilities $p_n(x_1^n)$ $=$
${\mathbf P}(X_1^n=x_1^n)$. A code (of length $N$) is a map
$x_1^n\in V^n\mapsto y_1^N\in W^N$ where $W$ is a `code alphabet'
(a typical example is where $V$ is the Latin alphabet
and $W$ $=$ $\{0,1\}$). One wants to have a code
of a shortest length $N(n)$ where
possible encoding errors
(when two different messages are identically encoded)
occur with asymptotically small probability. That is,
we want to extract a subset of messages $D_n\subset V^n$
of high probability $p_n(D_n)$ and low
cardinality $\#D_n$ (two opposite tasks).
We say that $h\geq 0$ is the
information rate of source $X_{-\infty}^\infty$ if
$\forall$ $R>h$ there exists a sequence $D_n\subset V^n$
such that $\#D_n\leq 2^{Rn}$ and $p_n(D_n)\to 1$; the value $h$
coincides with the Shannon (or Kolmogorov--Sinai) entropy
rate of $X_{-\infty}^\infty$. The shortest length $N(n)$
has $N(n)/n$ $\sim$ $h$.
In the talk, I will focus on a quantum analogue
of the coding problem.
I'll present various results in this direction, without assuming
any serious knowledge of quantum mechanics or classical
information theory.
Tuesday 14 May
2pm Michail Loulakis (ETH)
Einstein Relation for a tagged particle in the simple exclusion process
Consider the symmetric simple exclusion process, an interacting particle system
which is diffusive. If we tag a particle and follow its evolution, its position
converges after rescaling to a diffusion, characterized by a matrix D. Now, if
the tracer particle is driven by a small force, then it picks up a mean
velocity. The mean velocity is proportional to the small force and the constant
of proportionality, in the limit, can be computed from the diffusion matrix D
of the tracer particle with no driving force. This relationship, some times
referred to as the Einstein Relation, is believed to be generally valid. The
difficulty of the problem lies in the fact that the equilibria of the simple
exclusion process with a driven tracer particle are not generally accessible.
Using tools from large deviations, we establish the validity of Einstein
Relation for all symmetric simple exclusion processes in dimension greater
than or equal 3.
Tuesday 21 May (Joint with the Combinatorics seminar)
2pm Alan Stacey
Percolation on finite graphs
Tuesday 28 May
2pm John Kingman
Probability in battles
Tuesday 18 June
2pm Shiri Artstein (Tel Aviv)
Entropy increases at every step
Abstract. If X_i are IID random variables then the entropy
of
1/\sqrt n \sum_1^n X_i
increases with n.
Thursday 18 July
2pm N. Ganikhodjaev and U. Rozikov (Tashkent University
and Institute of Mathematics, Uzbekistan)
Gibbs
random fields on Cayley trees
Abstract.
We consider models of Gibbs random fields
on a tree: Ising, Potts, SOS and their generalizations.
The picture on a tree is quite different from that on
a lattice. For example, a ferromagnetic nearest
neighbour Ising model
with zero field on a tree has one translation-invariant
Gibbs measure for high temperatures and three at and below
the critical temperature (equal to
$J({\rm{arctanh}}1/k)^{-1}$ where $k$ is the number
of branches descending from a single vertex and $J>0$
a coupling constant). Besides, at low temperatures
there exist a continuum
of non-translation-invariant extreme Gibbs measures.
Similar results hold for the Potts model.
The talk will be made accessible without
any preliminary knowledge of Gibbs measure theory.
Graduate students especially are urged to attend.
Directions to the Laboratory.
The CMS is reached by a path along the east
side of the Isaac Newton Institute in Clarkson Road.
to return to Geoffrey Grimmett's home page.