Random Processes on Graphs, and Planar Brownian Motion

9803597佩雷斯首席研究员将在图上分析各种随机过程。例如，在统计力学、计算机科学和遗传重建中，噪声在树上传播的模型已被独立研究。只有当传播变量取两个值时，这个模型才能被很好地理解，当它可以用伊辛模型识别时。有两个不同的临界温度，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尔可夫态的纯度。波茨模型不知道第二个临界温度，在这种模型中，传播变量可以取两个以上的值。为了研究非树的大型同质图，研究者将应用最近与几位合作者共同开发的一种新的“质量传输方法”。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通过“分形渗透”构造的具有随机集的路径的交集等价。数学生物学家研究家谱中基因突变的传播，计算机科学家研究噪声通信网络中错误的传播，他们各自独立地得出了统计力学早期考虑的相同数学模型。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；其中一些涉及与计算机科学家的合作。最近出现了利用底层网络几何对称性的新方法，以更好地理解其上的随机过程；研究者将进一步发展和应用这些方法。最后，一个粒子的不规则随机运动的轨迹已经被概率学家深入研究；本研究将探讨这些痕迹的某些新方法，试图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.