On self-repellent one-dimensional random walks (Q582700)

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.