On the speed of the one-dimensional polymer in the large range regime

We consider a Hamiltonian involving the range of the simple random walk and the Wiener sausage so that the walk tends to stretch itself. This Hamiltonian can be easily extended to the multidimensional cases, since the Wiener sausage is well-defined in any dimension. In dimension one, we give a formula for the speed and the spread of the endpoint of the polymer path. Also, we provide the CLT. It can be easily showed that if the self-repelling strength is stronger, the end point is going away faster. This strict monotonicity of speed has not been proven in the literature for the one-dimensional case.