  We introduce a new disorder regime for directed polymers with one space and
one time dimension that is accessed by scaling the inverse temperature
parameter \beta with the length of the polymer n. We scale \beta_n := \beta
n^{-\alpha} for alpha non-negative. This scaling sits in between the usual weak
disorder (\beta = 0) and strong disorder regimes (\beta > 0). The fluctuation
exponents zeta for the polymer endpoint and \chi for the free energy depend on
\alpha in this regime, with \alpha = 0 corresponding to the usual polymer
exponents \zeta = 2/3, \chi = 1/3 and \alpha >= 1/4 corresponding to the simple
random walk exponents \zeta = 1/2, \chi = 0. For 0 < \alpha < 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.
