Transcript Prof. Dr. A. Walkolbinger and HD Dr. J.Geiger Seminary

Prof. Dr. A. Walkolbinger and HD Dr. J.Geiger
Seminary about Probablity Theory and related fields
Reporter: Mounir Balghouthi
Biased random walks on GaltonWatson trees
Introduction:

We consider random walks with a bias
toward the root on the family of tree T
of a supercritical galton-watson
branching process and we show that
the SPEED is POSITIVE whenever
the walk is transient, which is not the
case if the bias is directed away from
the root, the SPEED may be ZERO.
Fig.1

Choose some λ > 1, with probability
proportional to λ, the walker tries to walk in
direction of the root
The path of the walker:
On a different scale:
Expected number of visits to
a point:

Conditioned on having visited a point x , the
number of visits to x is distributed
geometrically
Fresh epoch:

Given a path <X0,X1,…>, we call n>0
a fresh epoch if Xn≠Xk for all k<n.
Regeneration epoch:

We call n>0 a regeneration epoch if n
is a fresh epoch and Xn+j ≠{X0,…,Xn-1};
for all j≥0.
Regeneration epochs:
Regeneration epochs on a
different scale:
Differences between
successive regeneration
epochs:
Idea of the proof:
1.
2.
3.
4.
5.
6.
Label the edges from each vertex y to its children by 1,…,d(y) so
each vertex is identified with the sequence of labels leading to it
from the root.
T is identified with a set [T] of finite sequences of positive
integers.
T(y) (the tree of descendants of y) is identified with the set [T(y)].
A path Y:=( Yk;k≥0) is descibed by the sequence of non negative
integersỶ:=(Ỷk;k≥1), where Ỷk is 0 if Yk is the parent of Yk-1and is
otherwise the label on the edge from Yk-1 to Yk .
Conditional on the event of non extinction, the sequence of fresh
trees T(Yζn) seen at regeneration epochs is stationary! But not
i.i.d. however, the part of tree between regeneration epochs,
together with the path taken through this part of tree is
independent of the rest of tree and the rest of walk. Call this part
Slabn.
The stationarity of the sequence of fresh trees seen at
regeneration epochs implies that the rs.vs. Slabn are identically
distributed.
References:
1.
1.
http://citeseer.ist.psu.edu/lyons96bia
sed.html
Russell Lyons, Rubin Pemantle &
Yuval Peres.
http://www.its.caltech.edu/~berger/bi
asedslides.pdf
Noam Berger, Nina Gantert & Yuval
Peres.