Topics in Percolation & Particle Models

该项目包括寻求回答概率论、相互作用粒子系统和渗透两个领域中出现的基本问题的研究。它们包括在（双）布朗网中向前和向后路径的接触点处的成核过程的构建和应用，以及在（临界）连续非简单环路过程的“双点”上的概念相关标记过程。这个循环过程是由pii和Federico Camia基于Schramm- loewner演化（SLE）和Schramm和Smirnov的工作构建的，是二维临界渗流的完整尺度限制。双点，即环自身和彼此接触的地方，代表了晶格点的尺度极限，微观状态的变化会对连通性产生宏观影响。在许多情况下，比如互联网的功能、工程材料的行为和经济，许多小个体（网站、分子、投资者）以看似随机的方式相互作用，导致整个系统的共同行为。概率论和统计学的一个经典目标就是理解这些情况。一组特别有趣的问题涉及所谓的“关键”系统，在这些系统中，即使一个人也可以为整个系统提供一个临界点。这个研究项目的目标是为理解这些关键系统所需的数学奠定坚实的基础，至少在一些特殊情况下，这些情况在过去几年中已经有了理论突破。希望这些特殊情况的进展将导致更普遍的进展。

This project comprises studies that seek to answer basic questions arising in two areas of Probability Theory, Interacting Particle Systems andPercolation. They include the construction and applicationof nucleation processestaking place at the touching points of forward and backwardpaths in the (double) Brownian web and the conceptually relatedmarking process on the ``double points'' of the (critical) continuumnonsimple loop process. This loop process, constructed by the PIand Federico Camia based on the Schramm-Loewner evolution (SLE)and the work of Schramm and Smirnov, is the full scaling limitfor two-dimensional critical percolation. The double points,where the loops touch themselves and each other, represent thescaling limits of lattice sites where a change in microscopic statehas a macroscopic effect on connectivity.There are many situations, such as the functioning of the Internetand the behavior of engineered materials and of the economy,where many small individuals (web sites, molecules, investors)interact with each other in seemingly random ways leading tonovel behavior of the entire system. A classic goal of ProbabilityTheory and Statistics is to understand these situations. A particularlyintriguing set of problems concerns so called ``critical'' systemswhere even one individual can provide a tipping point for the wholesystem. The goal of this research project is to lay a firm foundationfor the mathematics needed to understand such critical systems, at leastin some special situations where there have been theoretical breakthroughsin the past several years. It is hoped that progress in thesespecial cases will lead to more general progress.