Soft Polymers in Low Dimension

Abstract

In Chapters 3 and 4 we consider a variation of the SAW in which selfintersections are not forbidden but are penalized. We refer to this as a soft polymer. In Chapter 3 will show that the soft polymer has ballistic behavior in d = 1. The proof uses a Markovian representation of the local times of onedimensional SRW (a powerful technique that is useful also for other models), in combination with large deviation theory, variational calculus and spectral calculus. In Chapter 4 we will show that the soft polymer has diffusive behavior in d ? 5. The proof there uses a combinatorial expansion technique called the lace expansion, and is based on the idea that in high dimension SAW can be viewed as a “perturbation” of SRW. The above scaling says that in d = 1 and d ? 5 the soft polymer is in the same universality class as SAW. This is expected to be true also for 2 ? d ? 4, but a proof is missing. In Section 3.1 we define the model, in Section 3.2 we state the main result, a large deviation principle (LDP) for the location of the right endpoint. In Section 3.3 we outline a five-step program to prove the LDP for bridge polymers, i.e., polymers confined between their endpoints. This program is carried out in Section 3.4. In Section 3.5 we remove the bridge condition and prove the full LDP. It will turn out that the rate function has an interesting critical value strictly below the typical speed. The main technique that is used is the method of local times.