Department of Mathematics, University of Toronto, Toronto, Ontario, Canada.
Phys Rev Lett. 2010 Aug 27;105(9):090603. doi: 10.1103/PhysRevLett.105.090603. Epub 2010 Aug 24.
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.
我们通过将聚合物的长度 n 与逆温度β进行缩放，为 1+1 维的定向聚合物引入了一个新的无序状态。我们将β(n)≔βn(-α)进行缩放，其中α≥0。这种缩放在弱无序（β=0）和强无序（β>0）之间进行插值。在这个状态下，聚合物端点的涨落指数 ζ 和自由能的 χ 取决于α，其中α=0 对应于 Kardar-Parisi-Zhang 聚合物指数 ζ=2/3，χ=1/3，而α≥1/4 对应于简单随机游走指数 ζ=1/2，χ=0。对于α∈(0,1/4)，指数在这两个极端之间呈线性插值。在α=1/4 时，我们准确地确定了自由能的极限分布和聚合物的端点。
