Finite Markov chains in statistical mechanics and combinatorics

有限马尔可夫链可以看作是有限集合中的随机游动。在应用中，这个集合通常由某些组合对象组成，这些组合对象的典型属性有待理解。如果集合很大，获得这些问题的精确解通常是不可行的。在这种情况下，马尔可夫链可以提供一种高效的方法来生成随机近似，但前提是它们的平衡速度随着研究对象的大小而缓慢增长。该项目将研究几类马尔可夫链，这些马尔可夫链是为了研究统计力学和组合学中一些众所周知的困难问题而开发的，并确定在什么条件下它们提供有效的近似方案。

Finite Markov chains can be viewed as random walks in a finite set. In applications, this set often consists of certain combinatorial objects whose typical properties are to be understood. If the set is large, obtaining exact solutions to such problems is generally infeasible. Markov chains can provide a highly efficient method to generate randomised approximations in such cases, but only if they equilibrate at a rate that grows slowly with the size of the set of objects under study. The project will study several classes of Markov chains that have been developed to study a number of notoriously difficult problems in statistical mechanics and combinatorics, and determine under what conditions they provide efficient approximation schemes.