具有吸附和排斥作用的高分子的平衡态
On the approach to equilibrium for a polymer with adsorption and repulsion
摘要：
我们考虑一维简单随机游动的路径经过L步（L是偶数）后回到原点。在“钉扎模型”中，每条路径η具有权重λ^{N（η）}，其中λ>0并且N（η）是η中的零的数量。当路径被约束为非负时，聚合物被称为满足硬壁约束。众所周知，这种模型随着钉扎强度λ的变化而经历定域/离域转变。在本文中，我们研究了这些模型的自然“自旋翻转”动力学，并得到了几个估计的光谱间隙和混合时间。特别地，对于有壁的系统，我们证明了弛豫到平衡的速度总是至少与自由情况（λ=1，无壁）一样快，其中间隙和混合时间分别为L^{-2}和L ^2 \log L。这大大改善了以前已知的结果。对于没有壁的系统，我们证明了平衡相变具有清晰的动力学表现：对于λ \geq 1，弛豫至少与扩散自由的情况一样快，但在严格的离域相（λ< 1）中，间隙被证明为O（L^{-5/2}），直到对数校正。作为我们的界限的应用，我们证明了拉伸指数松弛的本地化制度的本地功能。
英文摘要：
We consider paths of a one-dimensional simple random walk conditioned to come back to the origin after L steps (L an even integer). In the 'pinning model' each path ηhas a weight λ^{N(η)}, where λ>0 and N(η) is the number of zeros in η. When the paths are constrained to be non-negative, the polymer is said to satisfy a hard-wall constraint. Such models are well known to undergo a localization/delocalization transition as the pinning strength λis varied. In this paper we study a natural 'spin flip' dynamics for these models and derive several estimates on its spectral gap and mixing time. In particular, for the system with the wall we prove that relaxation to equilibrium is always at least as fast as in the free case (λ=1, no wall), where the gap and the mixing time are known to scale as L^{-2} and L^2\log L, respectively. This improves considerably over previously known results. For the system without the wall we show that the equilibrium phase transition has a clear dynamical manifestation: for λ\geq 1 the relaxation is again at least as fast as the diffusive free case, but in the strictly delocalized phase (λ< 1) the gap is shown to be O(L^{-5/2}), up to logarithmic corrections. As an application of our bounds, we prove stretched exponential relaxation of local functions in the localized regime.
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