Elastic Polymers

Abstract

In this chapter we take a brief look at a version of the soft polymer where the penalty of the self-intersections decays with the loop length, i.e., the difference between the times at which the self-intersection occurs. This model is called the elastic polymer. Interestingly, it will turn out that this model has diffusive behavior in any d ? 1 as soon as the decay is sufficiently fast, namely, the critical dimension for diffusive behavior is lower than d = 4 and depends on the parameter controlling the rate of decay of the penalty as a function of the loop length. We will see that the scaling behavior of the elastic polymer is in fact highly sensitive to the value of this parameter. The lace expansion that was used in Chapter 4 can be carried over with minor modifications to arrive at the main scaling result. Section 5.1 defines the model, while Section 5.2 describes the main result, which is taken from van der Hofstad, den Hollander and Slade [162].