Trilogy of Couplings and General Formulas for Lower Bound of Spectral Gap

Abstract

This paper starts from a nice application of the coupling method to a traditional topic: the estimation of the spectral gap (=the first non-trivial eigenvalue). Some new variational formulas for the lower bound of the spectral gap of Laplacian on manifold or elliptic operators in Rd or Markov chains are reported [10],[15],[16]. The new formulas are especially powerful for the lower bounds; they have no common points with the classical variational formula (which goes back to Lord Rayleigh (1877) or E. Fischer (1905)) and is particularly useful for the upper bounds. No analog of the new formulas has appeared before. The formulas not only enable us to recover or improve the main known results but also make a global change of the study on the topic. This will be illustrated by comparison of the new results with the known ones in geometry. Next, we will explain the mathematical tools for proving the results. That is, the trilogy of the recent development of the coupling theory: the Markovian coupling, the optimal Markovian coupling and the construction of distances for coupling. Finally, some related results and some problems for further study are also mentioned. It is hoped that the paper could be readable not only for probabilists but also for geometers and analysts.