Average Free-Energy of Directed Polymers and Interfaces in Random Media

R. Acosta Diaz, C. D. Rodríguez-Camargo, N. F. Svaiter

Published 2016-09-22Version 1

We consider first the field theory formulation for a directed polymer in the presence of a quenched disorder. To obtain the average free-energy of this system we use the distributional zeta-function method. The distributional zeta-function is a complex function whose derivative at the origin yields the average free-energy of the system as the sum of two contributions: the first one is a series in which all the integer moments of the partition function of the model contribute; the second one, which can not be written as a series of the integer moments, but can be made as small as desired. Using the saddle-point equations of the model, and the replica symmetry ansatz, we obtain that the numbers of terms in the series representation for the average free-energy depends on the temperature of the system. Next, the field theory formulation of an interface in a quenched random potential is considered. Also using the results obtained from the distributional zeta-function method, the average free-energy of this system is presented. The $k$-th term of the series that defines the average free-energy represents an Euclidean field theory for a $k$-component Gaussian scalar field.