无序钉扎模型的复制耦合方法
A replica-coupling approach to disordered pinning models
摘要：
考虑更新过程τ={τ_0，τ_1，.}其中τ_i-τ_{i-1}的律有幂尾P（τ_i-τ_{i-1}=n）=n^{-（α+1）}L（n），α \ge 0，L（.）缓慢变化。然后，我们将一个随机的、n相关的奖励/惩罚分配给站点n属于tau的事件的发生。这类问题包括，除其他外，（1+d）-维模型的钉扎定向聚合物的一维随机缺陷，（1+1）-维模型的润湿无序基板，和波兰-Scheraga模型的DNA变性。通过改变奖励的平均值，系统经历了从τ占据N的有限分数的定域相到τ密度为零的非定域相的过渡。在没有无序的情况下，α>1的跃迁是一阶跃迁，α<1的跃迁是高阶跃迁。此外，当α从1到0变化时，跃迁的范围从一阶到无限阶.我们知道，只要α>1/2，即使是任意少量无序的存在也会改变跃迁的顺序。从物理学的角度来看，在这种情况下，无序是相关的，与启发式哈里斯标准一致。另一方面，对于0<α<1/2，最近K.亚历山大认为，如果无序足够弱，临界指数不会被随机性所改变：无序是无关紧要的。本文推广了自旋玻璃框架中的复制耦合和内插技术，给出了文[2]主要结果的一个新的、更简单的证明。此外，我们（部分）证明了[9]中对α<1/2的小无序展开，表明它提供了一个自由能上限，改善了退火的自由能上限。
英文摘要：
We consider a renewal process τ={τ_0,τ_1,...} on the integers, where the law of τ_i-τ_{i-1} has a power-like tail P(τ_i-τ_{i-1}=n)=n^{-(α+1)}L(n) with α\ge0 and L(.) slowly varying. We then assign a random, n-dependent reward/penalty to the occurrence of the event that the site n belongs to tau. This class of problems includes, among others, (1+d)-dimensional models of pinning of directed polymers on a one-dimensional random defect, (1+1)-dimensional models of wetting of disordered substrates, and the Poland-Scheraga model of DNA denaturation. By varying the average of the reward, the system undergoes a transition from a localized phase where τoccupies a finite fraction of N to a delocalized phase where the density of τvanishes. In absence of disorder the transition is of first order for α>1 and of higher order for α<1. Moreover, for αranging from 1 to 0, the transition ranges from first to infinite order. Presence of even an arbitrarily small amount of disorder is known to modify the order of transition as soon as α>1/2. In physical terms, disorder is relevant in this situation, in agreement with the heuristic Harris criterion. On the other hand, for 0<α<1/2 it has been proven recently by K. Alexander that, if disorder is sufficiently weak, critical exponents are not modified by randomness: disorder is irrelevant. In this work, generalizing techniques which in the framework of spin glasses are known as replica coupling and interpolation, we give a new, simpler proof of the main results of [2]. Moreover, we (partially) justify a small-disorder expansion worked out in [9] for α<1/2, showing that it provides a free energy upper bound which improves the annealed one.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
