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

We consider an ordinary one-dimensional recurrent random walk on $${\mathbb{Z}}$$. 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
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### References:

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