Quenched Path Measure and Pinning Effect

Abstract

In this chapter we present the proof of the pinning effect for the quenched path measure. This effect is related to a certain random variational problem which also governs the finer asymptotic behavior of the normalizing constant In Section 1 the pinning effect is discussed at a heuristic level. Section 2 shows how a certain weak pinning property can be bootstrapped into a strong pinning property. In Section 3 we introduce a simplified random variational problem in the spirit of Chapter 3 §3, and use it to introduce certain random length scales. Section 4 is devoted to the construction of a coarse graining method on the path space, which relies on the forest and clearings picture of Chapter 4. Finally Section 5 applies the coarse graining method of Section 4 to derive finer asymptotics on S _t,ω and establish a weak pinning property, which together with the results of Section 2 proves the full pinning effect.