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Biased random walks on Galton-Watson trees

Prof. Dr. A. Walkolbinger and HD Dr. J.Geiger Seminary about Probablity Theory and related fields Reporter: Mounir Balghouthi. Biased random walks on Galton-Watson trees. Introduction:.

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Biased random walks on Galton-Watson trees

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  1. Prof. Dr. A. Walkolbinger and HD Dr. J.Geiger Seminary about Probablity Theory andrelated fieldsReporter: Mounir Balghouthi Biased random walks on Galton-Watson trees

  2. 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.

  3. Fig.1 • Choose some λ > 1, with probability proportional to λ, the walker tries to walk in direction of the root

  4. The path of the walker:

  5. On a different scale:

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

  7. Fresh epoch: • Given a path <X0,X1,…>, we call n>0 a fresh epoch if Xn≠Xk for all k<n.

  8. 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.

  9. Regeneration epochs:

  10. Regeneration epochs on a different scale:

  11. Differences between successive regeneration epochs:

  12. Idea of the proof: • 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.

  13. References: • http://citeseer.ist.psu.edu/lyons96biased.html Russell Lyons, Rubin Pemantle & Yuval Peres. • http://www.its.caltech.edu/~berger/biasedslides.pdf Noam Berger, Nina Gantert & Yuval Peres.

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