Prof. Dr. A. Walkolbinger and HD Dr. J.Geiger Seminary about Probablity Theory and related fields Reporter: Mounir Balghouthi

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Prof. Dr. A. Walkolbinger and HD Dr. J.Geiger
Seminary about Probablity Theory and related
fieldsReporter Mounir Balghouthi

• Biased random walks on Galton-Watson trees

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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.

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Fig.1

• Choose some ? gt 1, with probability proportional
  to ?, the walker tries to walk in direction of
  the root

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The path of the walker
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On a different scale
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Expected number of visits to a point

• Conditioned on having visited a point x , the
  number of visits to x is distributed geometrically

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Fresh epoch

• Given a path ltX0,X1,gt, we call ngt0
• a fresh epoch if Xn?Xk for all kltn.

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Regeneration epoch

• We call ngt0 a regeneration epoch if n is a fresh
  epoch and Xnj ?X0,,Xn-1 for all j0.

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Regeneration epochs
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Regeneration epochs on a different scale
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Differences between successive regeneration
epochs
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Idea of the proof

1. 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.
2. T is identified with a set T of finite
   sequences of positive integers.
3. T(y) (the tree of descendants of y) is identified
   with the set T(y).
4. A path Y( Ykk0) is descibed by the sequence
   of non negative integers?(?kk1), where ?k is
   0 if Yk is the parent of Yk-1and is otherwise the
   label on the edge from Yk-1 to Yk .
5. Conditional on the event of non extinction, the
   sequence of fresh trees T(Y?n) 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.
6. The stationarity of the sequence of fresh trees
   seen at regeneration epochs implies that the
   rs.vs. Slabn are identically distributed.

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References

• http//citeseer.ist.psu.edu/lyons96biased.html
• Russell Lyons, Rubin Pemantle Yuval Peres.
• http//www.its.caltech.edu/berger/biasedslides.pd
  f
• Noam Berger, Nina Gantert Yuval Peres.