The intermediate disorder regime for directed polymers in dimension $1+1$
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 $β$ to zero as the polymer length $n$ tends to infinity. The natural choice of scaling is $β_n:=βn^{-1/4}$. 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 ($ζ=1/2,χ=0$), 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 $A_β$ that has the recently discovered crossover distributions as its one-point marginals, which for large $β$ 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.