Phase Transition: Questions in Percolation and Interacting Particle Systems

9704197利格特·艾伦·斯泰西建议研究渗透、相互作用粒子系统和概率组合学方面的问题。他提议（部分与汤姆·利格特合作）尝试扩展他们（和其他人）关于树上接触过程的工作：证明利格特的一个猜想，该猜想将完成对齐次树上接触过程的三个阶段的重要表征；更全面地了解三个阶段的行为；并利用这些详细的行为知识来完成与Robin Pemantle的工作，其中发现了某些指数增长的非均匀树，其中分支随机漫步和/或接触过程只有一个相变。Alan Stacey还提出扩展van den Berg开发的方法，以显示二维自旋玻璃模型的吉布斯测度在参数值上几乎肯定的唯一性，而对应的Ising模型的吉布斯测度不是唯一的。他建议研究渗透中的问题，如试图证明渗透发生在某些模型中，其中参数的任何非平凡值都是未知的，包括一个具有长期依赖关系的渗透模型，该模型源于图上随机行走的问题；一个更雄心勃勃的项目（鉴于许多杰出的概率学家的失败尝试）是证明在三维中标准键渗透的临界值处的渗透概率为零。Stacey还提出了一个吸引人的猜想，即在洗牌情况下产生的两个不同的马尔可夫链的收敛率是相同的。所描述的问题涉及重要的物理和生物过程的数学模型，例如疾病的传播和人口（包括人类和由其他生物组成的人口）的增长。尽管数学描述是抽象的，并且通常不是基于一个特定的现实世界的情况，但对抽象模型的详细理解有助于人们理解和预测重要的物理模型；事实上，一个或两个简单描述的数学模型可以与大量的物理系统相关。抽象模型和经验观察共有的一个关键现象是相变：一个系统在不同条件下可以表现出两种或两种以上完全不同的行为（或阶段）；一个经典的例子是，一种疾病可能在某些条件下传播失控，而在不同的情况下消失。从一个相到另一个相的变化通常是非常剧烈的，并且可能是系统所依赖的参数发生很小变化的结果，例如群体中个体的敏感性或温度（在另一个相变的经典例子中-物质从固体变为液体）。提出的工作探讨了相变的性质和阶段的类型，可以发生在许多不同的模型。它们包括接触过程——疾病传播的一种模式——在某些环境中可能表现为两个阶段，在其他环境中可能表现为三个阶段；而自旋玻璃——模拟含杂质物质的磁性——在某种情况下，人们认为不存在相变，系统在所有温度下都具有相似的行为；但这一猜想尚未得到证实。

9704197 Liggett Alan Stacey proposes to work on problems in percolation, interacting particle systems and probabilistic combinatorics. He proposes to work (partly with Tom Liggett) to try to extend work of theirs (and others) on contact processes on trees: to prove a conjecture of Liggett which would complete a significant characterization of the three phases of the process on a homogeneous tree; to understand more fully the behavior in the three phases; and to use this detailed knowledge of the behavior to complete work with Robin Pemantle in which certain exponentially growing non-homogeneous trees are found where the branching random walk and/or the contact process only have one phase transition. Alan Stacey also proposes to extend methods developed by van den Berg to show almost sure uniqueness of the Gibbs measure of a two-dimensional spin-glass model at parameter values where the Gibbs measure for the corresponding Ising model is not unique. He proposes to work on problems in percolation such as trying to prove that percolation occurs in certain models where this is not known for any non-trivial value of the parameters, including a percolation model with long-range dependencies which arises from a question about random walks on a graph; a more ambitious project (in the light of the unsuccessful attempts by many distinguished probabilists) is to show that the percolation probability at the critical value is zero for standard bond percolation in three dimensions. Stacey also proposes to work on an appealing conjecture that the rates of convergence are the same for two different Markov chains arising from a card-shuffling situation. The problems described concern mathematical models for important physical and biological processes such as the spread of disease and the growth of populations (both human, and those consisting of other organisms). Although the mathematical descriptions are abstract and are not typically based on one specific real-world situation, a detailed understandi ng of the abstract models helps one to understand and predict important physical model; indeed, one or two simply described mathematical models can be of relevance to a very large number of physical systems. One key phenomenon common to the abstract models and the empirical observations is phase transitions: a system can exhibit two or more quite different types of behavior (or phases) under different conditions; a classic example is that of a disease which may spread out of control under certain conditions and fizzle out under different circumstances. The change from one phase to another is often quite dramatic and can be the result of a very small change in the parameters on which the system depends, such as the susceptibility of the individuals in a population or (in another classic example of phase transition -- that of a substance turning from a solid to a liquid) the temperature. The work proposed explores the nature of the phase transition and the types of phases that can occur in a number of different models. They include the contact process -- a model for the spread of disease -- which can exhibit two phases in certain environments, and three phases in others; and spin glasses -- which model magnetism in a substance with impurities -- where, in a certain case, it is thought that no phase transition exists and the system has similar behavior at all temperatures; but this conjecture is not yet proven.