界面生长与种群动力学随机模型中的涨落与多重分形

中文摘要 AI 辅助

本论文研究无序与涨落在三个相关背景下如何塑造大规模生长现象：涨落界面、随机弹簧链和随机乘法增长。第一部分关注界面生长以及一种称为异常标度的现象，其中局部和全局指数存在差异。为阐明其起源，我们研究了两个模型中这种不匹配具有不同原因。第一个是异质弹性线，我们表明表观异常纯属统计效应。第二个是随机多孔介质方程，这是一个强非线性模型，其中异常是真实的。第二部分关注安德森局域化，它通过异质弹性线的谱性质出现在这里。我们研究了同时具有随机质量和随机弹簧常数的链，特别强调强无序区域，其中标准的弱无序展开失效。我们开发了一种新的组合方法来探测该区域。最后一部分关注具有再分配的随机乘法增长。在这类模型中，财富、人口或质量的变化与已有数量成正比，因此微小差异随时间放大。这自然产生宽分布，并为不平等和集中的出现提供了简单机制，应用于种群动力学、财富分配、城市增长和生态学。我们特别研究了每个位点具有自身淬火增长率（代表持续优势或劣势）且同时受瞬态涨落影响的模型。再分配通过均匀化系统与这些机制竞争。这种竞争导致丰富的相图。

英文摘要

This thesis studies how disorder and fluctuations shape large-scale growth phenomena in three related settings: fluctuating interfaces, random spring chains, and random multiplicative growth. The first part concerns interface growth and a phenomenon denoted anomalous scaling, where local and global exponents differ.To clarify its origin, we study two models in which this mismatch has different causes. The first is a heterogeneous elastic line, where we show that the apparent anomaly is a purely statistical effect. The second is the stochastic porous medium equation, a strongly nonlinear model in which the anomaly is genuine. The second part concerns Anderson localization, which appears here through the spectral properties of the heterogeneous elastic lines. We study chains with both random masses and random spring constants, with particular emphasis on the strong-disorder regime, where standard weak-disorder expansions break down. We develop a new combinatorial method to probe this regime. The last part concerns random multiplicative growth with redistribution. In such models, changes in wealth, population, or mass, for instance, are proportional to the amount already present, so that small differences are amplified over time. This naturally generates broad distributions and provides a simple mechanism for the emergence of inequality and concentration, with applications to population dynamics, wealth distribution, city growth, and ecology. We in particular study models in which each site has its own quenched growth rate, representing a persistent advantage or disadvantage, and is also subject to transient fluctuations. Redistribution competes with these mechanisms by homogenizing the system. This competition leads to rich phase diagrams.

发表机构

• LPENS（巴黎高等师范学院物理系）
• LPTMS（巴黎第十一大学理论与模型物理实验室）

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