Finite-temperature directed polymers in a random potential

The phase transition of directed polymers in a random potential has been studied by both weak-disorder expansions and numerical simulations. In dimension d=2+1, no evidence for a finite-temperature phase transition was found. Instead, the crossover length ${\mathit{t}}^{\mathrm{*}}$ is very large at high temperature T and has been found to vary as ln${\mathit{t}}^{\mathrm{*}}$\ensuremath{\sim}${\mathit{T}}^{2}$. In d=3+1, transfer-matrix studies show the existence of a phase transition and some of the critical exponents have been determined.