Intermediate Disorder Regime for Directed Polymers in Dimension 1+1

We introduce a new disorder regime for directed polymers in dimension 1 + 1 by scaling the inverse temperature beta with the length of the polymer n. We scale beta n : = beta n(-alpha) for alpha >= 0. This scaling interpolates between the weak disorder (beta = 0) and strong disorder regimes ( beta > 0). The fluctuation exponents zeta for the polymer end point and chi for the free energy depend on alpha in this regime, with alpha = 0 corresponding to the Kardar-Parisi-Zhang polymer exponents zeta = 2/3, chi = 1/3, 1= 3, and alpha >= 1/4 corresponding to the simple random walk exponents zeta = 1/2, chi = 0. For alpha is an element of (0, 1/4) the exponents interpolate linearly between these two extremes. At alpha = 1/4 we exactly identify the limiting distribution of the free energy and the end point of the polymer.