Reversible Markov chains with nonnegative spectrum

The title of the talk corresponds to a family of interesting random processes, which includes lazy random walks on graphs and much beyond them. Often, a key step in analysing such processes is to estimate their spectral gaps (ie. the difference between two largest eigenvalues). It is thus of interest to understand what else about the chain we can know from the spectral gap. We will present a simple comparison idea that often gives us the best possible estimates. In particular, we re-obtain and improve upon several known results on hitting, meeting, and intersection times; return probabilities; and concentration inequalities for time averages. We then specialize to the graph setting, and obtain sharp inequalities in that setting. This talk is based on work that has been in progress for far too long with Yuval Peres.