Aspects of Mass Transportation in Discrete Concentration Inequalities

Abstract

During the last half century there has been a resurgence of interest in Monge's 18th century mass transportation problem, with most of the activity limited to continuous spaces. This thesis, consequently, develops techniques based on mass transportation for the purpose of obtaining tight concentration inequalities in a discrete setting. Such inequalities on n-fold products of graphs, equipped with product measures, have been well investigated using combinatorial and probabilistic techniques, the most notable being martingale techniques. The emphasis here, is instead on the analytic viewpoint, with the precise contribution being as follows. We prove that the modified log-Sobolev inequality implies the transportation inequality in the first systematic comparison of the modified log-Sobolev inequality, the Poincar inequality, the transportation inequality, and a new variance transportation inequality. The duality shown by Bobkov and Gtze of the transportation inequality and a generating function inequality is then utilized in finding the asymptotically correct value of the subgaussian constant of a cycle, regardless of the parity of the length of the cycle. This result tensorizes to give a tight concentration inequality on the discrete torus. It is interesting in light of the fact that the corresponding vertex isoperimetric problem has remained open in the case of the odd torus for a number of years. We also show that the class of bounded degree expander graphs provides an answer, in the affirmative, to the question of whether there exists an infinite family of graphs for which the spread constant and the subgaussian constant differ by an order of magnitude. Finally, a candidate notion of a discrete Ricci curvature for finite Markov chains is given in terms of the time decay of the Wasserstein distance of the chain to its stationarity. It can be interpreted as a notion arising naturally from a standard coupling of Markov chains. Because of its natural definition, ease of calculation, and tensoring property, we conclude that it deserves further investigation and development. Overall, the thesis demonstrates the utility of using the mass transportation problem in discrete isoperimetric and functional inequalities.