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On self-repellent one-dimensional random walks. (English) Zbl 0691.60060

We consider an ordinary one-dimensional recurrent random walk on $ℤ$. A self-repellent random walk is defined by changing the ordinary law of the random walk in the following way: A path gets a new relative weight by multiplying the old one with a factor 1-$\beta$ for every self intersection of the path. $0<\beta <1$ is a parameter.

It is shown that if the jump distribution of the random walk has an exponential moment and if $\beta$ is small enough then the displacement of the endpoint is asymptotically of the order of the length of the path.

Reviewer: E.Bolthausen
##### MSC:
 60G50 Sums of independent random variables; random walks
##### Keywords:
self-repellent random walk; exponential moment
##### References:
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