Phase Transition: Questions in Percolation and Interacting Particle Systems

小行星9704197 Alan Stacey建议研究渗流，相互作用粒子系统和概率组合学中的问题。 他提议去工作（部分与汤姆利格特）试图延长他们的工作，（及其他）关于树上的接触过程：证明Liggett的一个猜想，该猜想将完成齐次树上接触过程的三个阶段的一个重要特征;更全面地理解这三个阶段的行为;并使用这种行为的详细知识来完成Robin Pemantle的工作，其中发现了某些指数增长的非齐次树，其中分支随机行走和/或或者接触过程只有一个相变。 艾伦·斯泰西还建议扩展货车登贝格开发的方法，以显示几乎肯定的唯一性的吉布斯措施的二维自旋玻璃模型的参数值，其中吉布斯措施相应的伊辛模型是不唯一的。 他建议工作的问题，在渗流，如试图证明，渗流发生在某些模型，这是不知道任何非平凡值的参数，包括一个渗流模型与长程依赖关系所产生的一个问题，关于随机游走图;一个更雄心勃勃的项目（鉴于许多杰出的probabilists不成功的尝试）是为了证明在临界值处的渗流概率对于三维的标准键渗流是零。 斯泰西还提出了一个有吸引力的猜想，即对于洗牌情况下产生的两个不同的马尔可夫链，收敛速度是相同的。 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（包括人类和其他生物体）的增长。 虽然数学描述是抽象的，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理解和预测重要的物理模型;实际上，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所共有的一个关键现象是相变：一个系统在不同的条件下可以表现出两种或更多种完全不同的行为（或阶段）;一个经典的例子是疾病可能在某些条件下失控，在不同的情况下失败。 从一个阶段到另一个阶段的变化往往是相当戏剧性的，并且可以是系统所依赖的参数的非常小的变化的结果，例如群体中个体的易感性或（在相变的另一个经典例子中-物质从固体变成液体）温度。提出的工作探讨了相变的性质和类型的阶段，可以发生在一些不同的模型。 它们包括接触过程-疾病传播的模型-在某些环境中可以表现出两个阶段，在其他环境中可以表现出三个阶段;和自旋玻璃-在含有杂质的物质中模拟磁性-在某种情况下，人们认为不存在相变，系统在所有温度下都有类似的行为;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.