Directed polymers in random media under confining force

Abstract

The scaling behavior of a directed polymer in a two-dimensional random potential under confining force is investigated. The energy of a polymer with configuration {y(x)} is given by H({y(x)}) = sigma(x=1)(N) eta(x,y(x)) + epsilonW(alpha), where eta(x,y) is an uncorrelated random potential and W is the width of the polymer. Using an energy argument, it is conjectured that the radius of gyration Rg(N) and the energy fluctuation deltaE(N) of the polymer of length N in the ground state increase as Rg(N) approximately N(nu) and deltaE(N) approximately N(omega), respectively, with nu = 1/(1+alpha) and omega = (1+2alpha)/(4+4alpha) for alpha > or = 1/2. An algorithm of finding the exact ground state, with the effective time complexity of O(N3), is introduced and used to confirm the conjecture numerically.