Numerical simulations for directed polymers in 1+1 and 2+1 dimensions

Numerical Simulations for directed polymers in 1 + 1 and 2 + 1 dimensions show that the transverse fluctuations and free energy fluctuations tend to the strong coupling limit: Deltax proportional tot(v), DeltaF proportional tot(v)(v = 2/3, omega = 1/3 for d = 1 + 1; v approximate to 0.6, omega approximate to 0.2 for d = 2 + 1) at any finite temperature as their behaviors at zero temperature for the long length t of polymer. At finite temperatures, as a result of that the specific heat in both 1 + 1 dimensions and 2 + 1 dimensions scales as t and the entropy fluctuations in 1 + 1 dimensions scale as t(1/2), the ensemble energy fluctuations in both 1 + 1 and 2 + 1 dimensions and the internal energy fluctuations in 1 + 1 dimensions are more pronounced than the free energy fluctuations and scale as t(1/2) There is an indication of a phase transition in 2 + 1 dimensions. It is plausible that the entropy fluctuations and the internal fluctuations per unit length polymers for t --> infinity will become zero value at high temperatures from nonzero values at low temperatures at some temperature T = T-C where the entropy fluctuation reaches maximum.