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

TL;DR

This paper analyzes the speed and spread of a one-dimensional polymer model influenced by a Hamiltonian involving the random walk's range and Wiener sausage, providing explicit formulas and monotonicity results.

Contribution

It derives a formula for the polymer's speed and spread in one dimension and proves the strict monotonicity of speed with respect to self-repelling strength.

Findings

Explicit formula for the polymer's speed in 1D

Formula for the spread of the endpoint

Monotonicity of speed with increasing self-repelling strength

Abstract

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. 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.