Random Processes on Graphs, and Planar Brownian Motion

9803597名首席研究员将分析图上的各种随机过程。例如，在统计力学、计算机科学和基因重建中，树木上的噪音传播模型已经被独立研究。只有当传播变量取两个值时，这个模型才能被很好地理解，当它可以与伊辛模型相一致时。存在两个截然不同的临界温度，一个是吉布斯态的唯一性，另一个是对应于自由边界条件的马氏态的纯洁性。对于Potts模型来说，第二个临界温度是未知的，在该模型中，传播变量可以取两个以上的值。为了研究不是树的大型同质图，研究人员将应用最近与几个合作者共同开发的一种新的“质量传输方法”。关于平面布朗路径的几何问题也将被研究；一个关键的工具将是路径与随机集的交集的相交等价，这些路径是通过“分形渗流”构造的。研究家谱中基因突变传播的数学生物学家和研究嘈杂通信网络中错误传播的计算机科学家，被独立地引向了统计力学中早先考虑的同一数学模型。首席研究人员将试图通过使用最新的概率技术来统一和扩展这些模型上的一些早期结果；其中一些涉及到与计算机科学家的合作。最近，出现了新的方法来利用底层网络几何中的对称性，以更好地理解网络上的随机过程；研究人员将进一步开发和应用这些方法。最后，粒子不规则随机运动的轨迹已经被概率学家深入研究；本研究将探索某些新的方法来研究这些轨迹，以尝试并阐明它们的几何结构。

9803597 Peres The principal investigator will analyze various random processes on graphs. For instance, a model of noisy propagation on trees has been studied independently in Statistical Mechanics, in Computer Science and in Genetic Reconstruction. This model is well understood only when the propagating variables take two values, when it can be identified with the Ising model. There are two distinct critical temperatures, one for uniqueness of Gibbs states and another for purity of the Markovian state that corresponds to free boundary conditions. The second critical temperature is not known for the Potts model, where the propagating variables can take more than two values. To study large homogeneous graphs that are not trees, the investigator will apply a new "mass transport method" developed recently with several collaborators. Problems on the geometry of the planar Brownian path will also be investigated; a key tool will be the intersection equivalence of the paths with random sets constructed via "fractal percolation". Mathematical biologists studying the spread of genetic mutations in a family tree, and computer scientists studying the propagation of errors in a noisy communication network, were led independently to the same mathematical model that was considered earlier in statistical mechanics. The principal investigator will attempt to unify and extend some of the earlier results on these models, by using recent probabilistic techniques; some of this involves collaboration with computer scientists. Recently, new methods have emerged to exploit symmetries in the geometry of the underlying network, to obtain better understanding of random processes on it; the investigator will develop and apply these methods further. Finally, the trace of the erratic random motion of a particle has been studied intensively by probabilists; this research will investigate certain new approaches to these traces to try and elucidate their geometric structure.