Random Processes on Graphs, and Planar Brownian Motion

小行星9803597 主要研究者将在图上分析各种随机过程。为 例如，树上的噪声传播模型已独立研究， 统计力学，计算机科学和遗传重建。该模型 只有当传播变量取两个值时才能很好地理解，当它可以 与伊辛模型一致。有两个不同的临界温度，一个是 吉布斯状态的唯一性和另一个对应的马尔可夫状态的纯度 自由边界条件。第二个临界温度是不知道的波茨 模型，其中传播变量可以取两个以上的值。学习大 齐次图不是树，研究人员将应用一个新的“质量运输”， 最近与几位合作者一起开发的“方法”。几何问题 平面布朗路径也将被研究;一个关键的工具将是相交 通过“分形渗流”构造的随机集的路径的等价性。 数学生物学家研究基因突变在家谱中的传播， 计算机科学家研究在嘈杂的通信中错误的传播 网络，被独立地引导到被认为是相同的数学模型， 在统计力学中。首席研究员将尝试统一和扩展 这些模型的一些早期结果，通过使用最近的概率技术;一些 这涉及到与计算机科学家的合作。最近，新的方法 它的出现是为了利用底层网络几何结构中的对称性， 更好地理解随机过程;研究人员将开发和应用 这些方法进一步。最后，一个粒子的不规则随机运动的轨迹 被概率学家深入研究;这项研究将调查某些新的 试图解释它们的几何结构。

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.