Monthus Cécile, Garel Thomas
Service de Physique Théorique, CEA/DSM/SPhT Unité de recherche associée au CNRS, Gif-sur-Yvette, France.
Phys Rev E Stat Nonlin Soft Matter Phys. 2006 May;73(5 Pt 2):056106. doi: 10.1103/PhysRevE.73.056106. Epub 2006 May 8.
We consider a directed polymer of length L in a random medium of space dimension d = 1,2,3. The statistics of low energy excitations as a function of their size l is numerically evaluated. These excitations can be divided into bulk and boundary excitations, with respective densities rho(bulk)(L) (E = 0,l) and rho(boundary)(L)(E=0,l). We find that both densities follow the scaling behavior rho(bulk, boundary)(L)(E = 0,l)=L(-1-theta)(d)R(bulk,boundary)(x = l/L), where theta(d) is the exponent governing the energy fluctuations at zero temperature (with the well-known exact value theta(1)= 1/3 in one dimension). In the limit x = l/L --> 0, both scaling functions R(bulk)(x) and R(boundary)(x) behave as R(bulk,boundary)(x) approximately x(-1-theta)(d), leading to the droplet power law rho(bulk, boundary)(L) (E = 0,l) approximately l(-1-theta)(d) in the regime 1 << l << L. Beyond their common singularity near x --> 0, the two scaling functions R(bulk,boundary)(x) are very different: whereas R(bulk)(x) decays monotonically for 0 < x < 1, the function R(boundary)(x) first decays for 0 < x < x(min), then grows for x(min) < x < 1, and finally presents a power law singularity R(boundary)(x) approximately (1-x)(-sigma)(d) near x -->1. The density of excitations of length l = L accordingly decays as rho(boundary)(L)(E = 0,l = L) approximately L(-lambda)(d) where gamma(d) = 1+ theta(d) - lambda(d). We obtain lambda(1) approximately 10.67, lambda(2) = 0.53, and lambda(3) approximately 0.39, suggesting the possible relation lambda(d) = 2theta(d).
我们考虑在空间维度d = 1、2、3的随机介质中长度为L的有向聚合物。对低能激发作为其尺寸l的函数的统计进行了数值评估。这些激发可分为体激发和边界激发，其各自的密度分别为rho(bulk)(L) (E = 0,l)和rho(boundary)(L)(E=0,l)。我们发现，这两种密度都遵循标度行为rho(bulk, boundary)(L)(E = 0,l)=L(-1-theta)(d)R(bulk,boundary)(x = l/L)，其中theta(d)是控制零温度下能量涨落的指数（在一维中具有著名的精确值theta(1)= 1/3）。在x = l/L --> 0的极限情况下，两个标度函数R(bulk)(x)和R(boundary)(x)的行为均为R(bulk,boundary)(x)近似为x(-1-theta)(d)，从而在1 << l << L的区域中导致液滴幂律rho(bulk, boundary)(L) (E = 0,l)近似为l(-1-theta)(d)。除了在x --> 0附近的共同奇点外，两个标度函数R(bulk,boundary)(x)非常不同：R(bulk)(x)在0 < x < 1时单调衰减，而函数R(boundary)(x)在0 < x < x(min)时先衰减，在x(min) < x < 1时增长，最后在x -->1附近呈现幂律奇点R(boundary)(x)近似为(1-x)(-sigma)(d)。长度l = L的激发密度相应地衰减为rho(boundary)(L)(E = 0,l = L)近似为L(-lambda)(d)，其中gamma(d) = 1+ theta(d) - lambda(d)。我们得到lambda(1)近似为10.67，lambda(2) = 0.53，lambda(3)近似为0.39，这表明可能存在关系lambda(d) = 2theta(d)。
