On self-repellent one-dimensional random walks

Bolthausen, E (1990). On self-repellent one-dimensional random walks. Probability Theory and Related Fields, 86(4):423-441.

Abstract

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-β for every self intersection of the path. 0<β<1 is a parameter.
It is shown that if the jump distribution of the random walk has an exponential moment and if β is small enough then the displacement of the endpoint is asymptotically of the order of the length of the path.

Abstract

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-β for every self intersection of the path. 0<β<1 is a parameter.
It is shown that if the jump distribution of the random walk has an exponential moment and if β is small enough then the displacement of the endpoint is asymptotically of the order of the length of the path.

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Citations

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