Random Discrete Structures

小行星9803249 PI将研究概率论与 离散数学 其中一个问题本质上是组合问题， 与枚举的格路径和解决一类递归中出现的， 问题.这个问题源于计算机科学中的一个应用程序，但应用程序 这一组合分析扩展到粒子系统，以及许多其他枚举 问题第二个研究领域是可数随机过程理论 组 许多基本过程，如渗流，随机游走，马尔可夫随机 场相对于某个群的作用是对称的。 当物体的几何形状 空间是双曲的，群通常是不顺从的，过程的对称性 在这样一个大集团的作用下， 表现良好 PI将尝试利用一些最近开发的质量传输方法 为了获得这样的结果。最后，他将继续研究树上的随机过程， 特别是转子模型的检查、接触过程、分支 随机游走和入侵渗透 通常数学和物理意义上的数量都被编码在一个幂中 系列，这是一个非常强大的工具，用于获得表达式的数量 感兴趣，只要最终能解码信息。 这是众所周知的 对于一个变量的幂级数，但当有多个变量时， 变量 即使是合理的近似值一般也是未知的。 结果是 研究预计将包括一个可靠的方法，近似系数 多变量幂级数这将允许近似解， 问题的范围从路径枚举，排队问题（计算效率 生产网络），分析算法的速度（最初的动机是 这个项目）。

9803249 Pemantle The PI will study several problems on the interface between probability theory and discrete mathematics. One of these problems is essentially combinatorial, having to do with enumeration of lattice paths and with solving a class of recursions arising in this problem. The problem arose from an application in computer science, but the applications of this combinatorial analysis extend to particle systems, and to many other enumeration problems. The second research area is the theory of random processes on countable groups. Many basic processes such as percolation, random walk, and Markov random fields are symmetric with respect to the action of some group. When the geometry of the space is hyperbolic, the group is typically non-amenable, and the symmetry of the process under the action of such a large group forces the process to be at least somewhat well behaved. The PI will attempt to exploit some recently developed mass transport methods to obtain such results. Finally, he will continue research into random processes on trees, and in particular, the examination of the rotor model, the contact process, branching random walk, and invasion percolation. Often quantities of mathematical and physical interest are encoded in a power series, which is a very powerful tool for obtaining expressions for the quantities of interest, provided one can in the end decode the information. It is known how to do this for power series in one variable, but relatively little is known when there is more than one variable. Even a reasonable approximation is in general unknown. The results from this research are expected to include a reliable method for approximating the coefficients of power series in more than one variable. This will then allow approximate solutions to problems that range from path enumeration, to queuing problems (computing efficiency of networks of production), to analysis of the speed of algorithms (the original motivation for this project).