  We introduce a new disorder regime for directed polymers in dimension $1+1$
that sits between the weak and strong disorder regimes. We call it the
intermediate disorder regime. It is accessed by scaling the inverse temperature
parameter $\beta$ to zero as the polymer length $n$ tends to infinity. The
natural choice of scaling is $\beta_n:=\beta n^{-1/4}$. We show that the
polymer measure under this scaling has previously unseen behavior. While the
fluctuation exponents of the polymer endpoint and the log partition function
are identical to those for simple random walk ($\zeta=1/2,\chi=0$), the
fluctuations themselves are different. These fluctuations are still influenced
by the random environment, and there is no self-averaging of the polymer
measure. In particular, the random distribution of the polymer endpoint
converges in law (under a diffusive scaling of space) to a random absolutely
continuous measure on the real line. The randomness of the measure is inherited
from a stationary process $A_{\beta}$ that has the recently discovered
crossover distributions as its one-point marginals, which for large $\beta$
become the GUE Tracy-Widom distribution. We also prove existence of a limiting
law for the four-parameter field of polymer transition probabilities that can
be described by the stochastic heat equation. In particular, in this weak noise
limit, we obtain the convergence of the point-to-point free energy fluctuations
to the GUE Tracy-Widom distribution. We emphasize that the scaling behaviour
obtained is universal and does not depend on the law of the disorder.
