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 β with the length of the polymer n. We scale β(n)≔βn(-α) for α≥0. This scaling interpolates between the weak disorder (β=0) and strong disorder regimes (β>0). The fluctuation exponents ζ for the polymer end point and χ for the free energy depend on α in this regime, with α=0 corresponding to the Kardar-Parisi-Zhang polymer exponents ζ=2/3, χ=1/3, and α≥1/4 corresponding to the simple random walk exponents ζ=1/2, χ=0. For α∈(0,1/4) the exponents interpolate linearly between these two extremes. At α=1/4 we exactly identify the limiting distribution of the free energy and the end point of the polymer.