Localization of directed polymers in random media due to a columnar defect

Abstract

We investigate the localization of directed polymers in random media due to the presence of an attractive columnar defect at the center of a two-dimensional substrate. If the defect’s strength is too weak to affect the polymers, the localization length of the polymers exhibits a power-law behavior as a function of the polymer length, as in the case of no defect. When the defect’s strength is greater than a critical value, the localization length approaches a finite value, thereby yielding a localization length exponent (or liberation exponent) ν ⊥ = 1.8004(31). The correlation length, which is perpendicular to the localization length, is defined as the distance over which the polymer is localized by the defect. The correlation length exponent ν ‖ = 2.881(5) is estimated from the data collapse via the scaling relation ζ = ν ⊥/ν ‖, where ζ = 5/8 represents the wandering exponent. In addition, we measure the number of times that the optimal path passes through the defect, because that number increases as the defect’s strength increases. This measurement yields a new critical exponent λ = 0.59.