Walks in Random Media, Stochastic Growth and Pinning Effects.

随机介质游动、随机增长和钉扎效应。

• 批准号: EP/L012154/1
• 负责人: Nikolaos Zygouras
• 金额: $ 39.28万
• 依托单位: University of Warwick
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2014
• 资助国家: 英国
• 起止时间: 2014 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FL012154%2F1
• 关键词: Walks; Random; Media; Stochastic; Growth

A cornerstone of probability theory has been the establishment of the Law of Large Numbers and the Central Limit Theorem, both of them having impact beyond mathematical sciences. Roughly speaking, the sum of n independent, identically distributed variables with finite second moment is macroscopically of order n and has fluctuations of order n^{1/2}, obeying the Gaussian distribution. Underlying the Gaussian fluctuations is the linear dependence of the sum on the collection of the independent variables. However, most phenomena in nature exhibit a nonlinear dependence on the inherent randomness and the challenge is to (i) understand the nonlinear structure that propagates the randomness and (ii) reveal the universal features of this mechanism.Random walks in random media are widely used to model such phenomena in statistical physics. Two such instances that are receiving increasingly high attention are (A) stochastic growth models and (B) pinning models on defect lines. In case (A) one deals with a randomly growing interface. The non rigorous work of Kardar-Parisi-Zhang (KPZ) in the mid 80's set the framework of what is currently known as the KPZ universality class, by predicting that this class of models exhibits t^{1/3} fluctuations. More recent mathematical works have related, in special cases, the fluctuations of such systems to those coming from the theory of random matrices. Our goal is to build a rigorous mathematical theory that will explain the nature of these fluctuations by looking into the exactly solvable nature of these models, connect it to other mathematical fields and eventually perturb it in order to reveal universal phenomena.In case (B) one deals with a random walk in the vicinity of a defect line. The goal is to understand phase transitions related to localization and delocalization phenomena. Techniques related to large deviations and coarse graining have been used recently to study the phase diagrams of such phenomena. While progress has been made a number of important questions remain unresolved.We propose to provide a new path in the field through the construction of continuum limits of such models. In this way we aim to resolve the open questions and also make deep and novel connections to KPZ phenomena.

概率论的一个基石是大数定律和中心极限定理的建立，它们的影响都超出了数学科学。粗略地说，具有有限二阶矩的n个独立同分布变量的和在宏观上是n阶的，并且具有n^{1/2}阶的波动，服从高斯分布。高斯波动的基础是和对自变量集合的线性依赖。然而，自然界中的大多数现象都表现出对固有随机性的非线性依赖，挑战在于（i）理解传播随机性的非线性结构以及（ii）揭示这种机制的普遍特征。随机介质中的随机游走被广泛用于统计物理中对此类现象进行建模。两个这样的例子，正受到越来越高的关注是（A）随机增长模型和（B）钉扎模型的缺陷线。在情形（A）中，我们处理的是一个随机增长的界面。Kardar-Parisi-Zhang（KPZ）在80年代中期的非严格工作，通过预测这类模型表现出t^{1/3}涨落，建立了目前称为KPZ普适类的框架。最近的数学著作在特殊情况下将这种系统的涨落与随机矩阵理论的涨落联系起来。我们的目标是建立一个严格的数学理论，通过研究这些模型的精确可解性质来解释这些波动的性质，将其与其他数学领域联系起来，并最终扰动B，以揭示普遍现象。在情况（B）中，我们处理缺陷线附近的随机行走。目标是了解与定域和离域现象有关的相变。大偏差和粗粒化的相关技术最近已被用来研究这种现象的相图。虽然已经取得了一些进展，一些重要的问题仍然没有得到解决，我们建议在该领域提供一个新的路径，通过建设连续的限制，这样的模型。通过这种方式，我们的目标是解决悬而未决的问题，并与KPZ现象建立深刻而新颖的联系。

项目成果

A penalised model reproducing the mod-Poisson fluctuations in the Sathe-Selberg theorem

再现 Sathe-Selberg 定理中模泊松涨落的惩罚模型

• DOI: 10.48550/arxiv.1701.03432
• 发表时间: 2017
• 期刊: arXiv e-prints
• 作者: Barhoumi-Andr

Bivariate fluctuations for the number of arithmetic progressions in random sets

随机集中算术级数数的双变量波动

• DOI: 10.1214/19-ejp391
• 发表时间: 2019
• 期刊: Electronic Journal of Probability
• 影响因子: 1.4
• 作者: Barhoumi-Andréani Y

Random Characters under the $L$-Measure, I: Dirichlet Characters

$L$-Measure 下的随机字符，I：狄利克雷字符

• DOI: 10.1093/imrn/rnx168
• 发表时间: 2019
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Barhoumi-Andréani Y

Polynomial chaos and scaling limits of disordered systems

• DOI: 10.4171/jems/660
• 发表时间: 2013-12
• 期刊: Journal of the European Mathematical Society
• 影响因子: 2.6
• 作者: F. Caravenna;Rongfeng Sun;Nikos Zygouras

Point-to-line polymers and orthogonal Whittaker functions

点到线聚合物和正交 Whittaker 函数

• DOI: 10.48550/arxiv.1703.07337
• 发表时间: 2017
• 期刊: arXiv e-prints
• 作者: Bisi Elia