This article investigates the effect for random pinning models of long range power-law decaying correlations in the environment. For a particular type of environment based on a renewal construction, we are able to sharply describe the phase transition from the delocalized phase to the localized one, giving the critical exponent for the (quenched) free-energy, and proving that at the critical point the trajectories are fully delocalized. These results contrast with what happens both for the pure model (i.e. without disorder) and for the widely studied case of i.i.d. disorder, where the relevance or irrelevance of disorder on the critical properties is decided via the so-called Harris Criterion.
Pinning on a defect line: characterization of marginal disorder relevance and sharp asymptotics for the critical point shift
钉扎模型中的无序效应是理论物理和严格数学物理中备受关注的课题。一个特殊的兴趣点是淬火和退火的临界点，少量的无序重合的问题。在过去的几年里，这个问题在大多数情况下都得到了数学上的解决，特别是对哈里斯标准的严格验证。然而，边际情况，其中返回概率指数等于$1/2$，即其中更新过程的到达间隔定律由$K(n)=n^{-3/2}ϕ(n)$给出，其中$ϕ$是缓慢变化的函数，已经被部分开放。在本文中，我们给出了一个完整的答案，通过证明一个简单的必要和充分的标准，返回概率的无序相关性，这证实了早期的预测文献。此外，我们还给出了临界点移动的尖锐渐近性：在一维简单随机游动的钉扎（或润湿）情形下，临界点的移动满足下列高温渐近性：$$ \lim_{β\rightarrow 0}β^2\log h_c(β)= - \fracπ{2}. $$这对B的一个主张给出了严格的证明. Derrida，V. Hakim and J. Vannimenus（Journal of Statistical Physics，1992）.
Scaling limit for the pinning model in correlated Gaussian environment beyond the $L^2$-regime
Research Center for Mathematics and Interdisciplinary Sciences, Shandong University（山东大学数学与交叉科学研究中心）；Center for Applied Mathematics, Tianjin University（天津大学应用数学中心）；Department of Financial and Actuarial Mathematics, Xi’an Jiaotong-Liverpool University（西交利物浦大学金融精算数学系）
