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Attracting edge property for a class of reinforced random walks. (English) Zbl 1057.60048
The author considers the so-called edge-reinforced random walk with weight function $$W$$ on a connected (unoriented) graph of bounded degree. That is, the walker jumps along the edges, and the probability along a given edge is given by the quotient of the value of $$W$$ of the number of previous jumps along that edge, divided by the sum of all the $$W$$-values of the number of previous jumps along all the adjacent edges of the present site. Here $$W$$ is a positive weight function, which is assumed to be $$W(k)=k^\rho$$ for some $$\rho>1$$.
It was previously known that, for the graph $$\mathbb Z^ d$$, under the condition $$\sum_{k\in\mathbb N}1/W(k)<\infty$$, there is almost surely one (random) attracting edge which is traversed infinitely often. The present paper strengthens this result for arbitrary bounded-degree graphs with $$W(k)=k^\rho$$ for some $$\rho>1$$, by showing that this edge is unique, almost surely. The main tools are martingale techniques and a comparison to the generalized urn scheme. The assumption that $$W(k)=k^\rho$$ is crucial for most of the arguments.

##### MSC:
 60G50 Sums of independent random variables; random walks
##### Keywords:
reinforced random walk; supermartingale; coupling; urn
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##### References:
  ATHREy A, K. B. and KARLIN, S. (1986). Embedding of Urn schemes into continuous time Markov branching processes and related limit theorems. Ann. Math. Statist. 39 1801- 1817. · Zbl 0185.46103 · doi:10.1214/aoms/1177698013  DAVIS, B. (1990). Reinforced random walk. Probab. Theory Random Fields 84 203-229. · Zbl 0665.60077 · doi:10.1007/BF01197845  DIACONIS, P. (1988). Recent progress on de Finetti’s notion of exchangeability. In Bayesian Statistics 3 (J. M. Bernardo, M. H. DeGroot, D. V. Lindley and A. F. M. Smith, eds.) 43-110. Oxford Univ. Press. · Zbl 0707.60033  DURRETT, R. (1991). Probability: Theory and Examples. Wadsworth and Brooks/Cole, Belmont, CA. · Zbl 0709.60002  DURRETT, R., KESTEN, H. and LIMIC, V. (2002). Once edge-reinforced random walk on a tree. Probab. Theory Related Fields 122 567-592. · Zbl 0995.60042 · doi:10.1007/s004400100179  HILL, B. M., LANE, D. and SUDDERTH, W. A. (1980). Strong law for some generalized urn processes. Ann. Probab. 8 214-226. · Zbl 0429.60021 · doi:10.1214/aop/1176994772  KEANE, M. S. and ROLLES, S. W. W. (2000). Edge-reinforced random walk on finite graphs. In Infinite Dimensional Stochastic Analy sis (P. Clement, F. den Hollander, J. van Neerven and B. de Pagter, eds.) 217-234. Roy al Netherlands Academy of Arts and Sciences, Amsterdam. · Zbl 0986.05092  PEMANTLE, R. (1988). Phase transition in reinforced random walk and RWRE on trees. Ann. Probab. 16 1229-1241. · Zbl 0648.60077 · doi:10.1214/aop/1176991687  PEMANTLE, R. (1992). Vertex-reinforced random walk. Probab. Theory Related Fields 92 117-136. · Zbl 0741.60029 · doi:10.1007/BF01205239  PEMANTLE, R. (2001). Random processes with reinforcement.  PEMANTLE, R. and VOLKOV, S. (1999). Vertex-reinforced random walk on Z has finite range. Ann. Probab. 27 1368-1388. · Zbl 0960.60041 · doi:10.1214/aop/1022677452  SELLKE, T. (1994). Reinforced random walks on the d-dimensional integer lattice. · Zbl 1154.82011  TARRÈS, P. (2001). VRRW on Z eventually gets stuck at a set of five points.  TÓTH, B. (1997). Limit theorems for weakly reinforced random walks on Z. Studia Sci. Math. Hungar. 33 321-337. · Zbl 0912.60043  VOLKOV, S. (2001). Vertex-reinforced random walk on arbitrary graphs. Ann. Probab. 29 66-91. · Zbl 1031.60089 · doi:10.1214/aop/1008956322  VANCOUVER, BRITISH COLUMBIA CANADA V6T 1Z2 E-MAIL: limic@math.ubc.ca
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