ATTRACTING EDGE AND STRONGLY EDGE REINFORCED WALKS
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Vlada Limic and Pierre Tarres
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The goal is to show that an edge reinforced random        
walk on a graph of bounded degree, with          
reinforcement {\em weight function} $W$ taken from a general class of         
reciprocally summable reinforcement weight functions, traverses a      
random {\em attracting} edge at all large times.         
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The statement of the main theorem is very close to settling          
a conjecture of Sellke (1994).          
An important corollary of this main result says that          
if $W$ is reciprocally summable and nondecreasing,         
the attracting edge exists on any graph of bounded degree, with          
probability $1$.         
Another corollary is the main theorem of Limic (2003) where         
the class of weights was restricted to reciprocally summable powers. 
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The proof uses martingale and other techniques developed          
by the authors in separate studies of edge and vertex reinforced          
walks (Limic (2003), Tarr\`es (2004)), 
and of nonconvergence properties of stochastic algorithms towards          
unstable equilibrium points of the associated deterministic dynamics,
Tarr\`es (2000).