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Vertex-reinforced random walk
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  • Published: March 1992

Vertex-reinforced random walk

  • Robin Pemantle1 

Probability Theory and Related Fields volume 92, pages 117–136 (1992)Cite this article

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Summary

This paper considers a class of non-Markovian discrete-time random processes on a finite state space {1,...,d}. The transition probabilities at each time are influenced by the number of times each state has been visited and by a fixed a priori likelihood matrix,R, which is real, symmetric and nonnegative. LetS i (n) keep track of the number of visits to statei up to timen, and form the fractional occupation vector,V(n), where\(v_i (n) = {{S_i (n)} \mathord{\left/ {\vphantom {{S_i (n)} {\left( {\sum\limits_{j = 1}^d {S_j (n)} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\sum\limits_{j = 1}^d {S_j (n)} } \right)}}\). It is shown thatV(n) converges to to a set of critical points for the quadratic formH with matrixR, and that under nondegeneracy conditions onR, there is a finite set of points such that with probability one,V(n)→p for somep in the set. There may be more than onep in this set for whichP(V(n)→p)>0. On the other handP(V(n)→p)=0 wheneverp fails in a strong enough sense to be maximum forH.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin, 53706, Madison, WI, USA

    Robin Pemantle

Authors
  1. Robin Pemantle
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Additional information

This research was supported by an NSF graduate fellowship and by an NSF postdoctoral fellowship

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Cite this article

Pemantle, R. Vertex-reinforced random walk. Probab. Th. Rel. Fields 92, 117–136 (1992). https://doi.org/10.1007/BF01205239

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  • Received: 27 September 1989

  • Revised: 18 November 1991

  • Issue Date: March 1992

  • DOI: https://doi.org/10.1007/BF01205239

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Keywords

  • State Space
  • Stochastic Process
  • Random Walk
  • Probability Theory
  • Random Process
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