abstract:
We will discuss two different classes of self-interacting random walks. Firstly, in joint work with Akira Sakai, we consider a simple model of a random walk with reinforcement but with very short term spatial memory. The simplicity of these "Senile random walks" enables us to prove many things such as recurrence/transience and an exact expression for the diffusion constant. Secondly, in joint work with Remco van der Hofstad, we derive an expansion for general self-interacting random walks. We use the expansion to prove a central limit theorem for a class of once reinforced random walks with non-zero drift (all dimensions) and for excited random walk (high dimensions), when the reinforcement and excitement parameters are sufficiently small.
abstract:
In this problem session, I will
discuss some interesting open problems in the three subjects mentioned in
the title. Of course, I will give some background as well, so the session
can be followed even if you have never worked in any of these areas. Some
keywords are: stationary distributions for sandpiles, critical values for
Bak-Sneppen, scaling limits, phase diagrams and "p_c=1/2?".
Summary of the problems discussed
1. r_c=1/2.
Take a subcritical random cluster measure with parameters p and q, and
choose a parameter 0<r<1. We colour echt cluster red with probability r, and
green with probability 1-r, independent of each other. Question: what is the
critical value for r, that is, for what values of r do we get infinite red
components?
For q=1 (q is the parameter of the random cluster measure) the answer is
that r_c=1/2, and this does not depend on p. This is recent work by Balint,
Camia and Meester (in preparation). We conjecture that also for other values
of q, the critical value r_c is equal to 1/2. In particular the case q=2 is
interesting, since for q=2, and r=1/2 we obtain the ising model.
Reference: Olle Haggstrom, coloring percolation clusters at random,
Stoch. Proc. Rel. Fields.
2. Scaling limits for fractal percolation.
In fractal percolation we have two parameters p and N. p is the retention
probability (as usual) and NxN is the number of squares in which we
subdivide each retained square for the next iteration. When we send N to
infinity, the model behaves more and more like independent site percolation.
(see Falconer and Grimmett) Instead of squares, we can also do fractal
percolation on triangles. It is then natural to conjecture that the scaling
limit, as N tends to infinity, and p=1/2, is the same as the scaling limit
for independent site
percolation on the triangular lattice. This would be an interesting and
non-trivial example of universality.
Reference: Falconer and Grimmett, On the geometry of random cantor
sets and fractal percolation. Journal of Theor. Prob.
3. SNIR-percolation
Consider a variant of the usual continuum percolation model: start with a
Poisson process of density d, and connect to points x and y of the process
if and only if SNIR(x-y) T and SNIR(y-x) T, where T is some threshold and
SNIR(x-y) is the ratio of l(x,y) and 1 + j\sum_z l(z,y), where l is some
real function satisfying some natural conditions (like being a function of
the distance only, and being non-increasing in this distance), and where the
sum is over all Poisson points z, and where j0 is a parameter.
We would like to understand the two-dimensional phase-diagram in the density
d and paramater j 0. Partial results are know (see reference below) but the
lack of monotonicity makes analysis difficult.
Reference: Dousse, Franceschetti, Macris, Meester, Thiran,
Percolation in the signal to noise interference ratio graph, to appear in J.
Appl. Prob.
(see also
homepage of the speaker)