Directed polymers and randomness

Abstract

The problem of a directed polymer in a random medium on hierarchical lattices has been considered. Randomness is introduced by allowing the interaction energy to be random on each and every bond. Several features including a generalization of the Harris criterion for this criticality via relevant disorder and aspects of unzipping have been discussed. This chapter discusses the effects of two types of randomness on the behavior of directed polymers. The significance of directed polymer lies in the fact that the pure system is very well understood and exactly solvable in all dimensions while the random problem can be attacked in several different ways, a luxury not affordable in most situations. Two types of randomness can arise in the context of directed polymers. One type would involve imposition of a random external potential (random medium problem). In the second type, the interaction (between say two chains) is random (RANI model). In the random medium problem, the random potential would like to have a realization dependent ground state which may not match with the zero-field state. In the RANI model, the randomness in the interaction may lead to a change in the phase transition behavior shown by the polymers. In both cases, disorder turns out to be marginally relevant though at two different dimensions. The random medium problem is better understood compared to the RANI model.