Duality and evolving set bounds on mixing times

We sharpen the Evolving set methodology of Morris and Peres and extend it to study convergence in total variation, relative entropy, L2 and other distances. Bounds in terms of a modified form of conductance are given which apply even for walks with no holding probability. These bounds are found to be strictly better than earlier Evolving set bounds, may be substantially better than conductance profile results derived via Spectral profile, drastically sharpen Blocking Conductance bounds if there are no bottlenecks at small sets, and give intuition into the workings of Canonical Path methods. This paper is intended solely to develop theoretical underpinnings, and as such we focus on two points : proving the sharpest most general results we can, and showing the Evolving Set methods to be better than previous isoperimetric methods. In order to learn about Evolving Sets we recommend the relevant chapter in our book with Tetali [23], and of course the original paper of Morris and Peres [26]. To learn about some applications please see our paper on Cheeger Inequalities [21], that on Canonical Path bounds for non-lazy walks [18], our alternate interpretation of Morris’ study of the Thorp shuffle [23, 25], Morris’ paper on the Exclusion process [24], and the paper of Diaconis and Fill on the duality method [2].