Concentration on the Discrete Torus Using Transportation

Source abstract

The sub-Gaussian constant of a graph arises naturally in bounding the moment-generating function of Lipschitz functions on the graph, with a given probability measure on the set of vertices. The closely related spread constant of a graph measures the maximal variance over all Lipschitz functions on the graph. As such they are both useful (as demonstrated in the works of Bobkov, Houdré and Tetali and Alon, Boppana and Spencer) for describing the concentration of measure phenomenon in product graphs. An equivalent formulation of the sub-Gaussian constant using a transportation inequality, introduced by Bobkov and Götze, is investigated here in depth, leading to a new way of bounding the sub-Gaussian constant. A tight concentration result for the discrete torus is given as a concrete application. An infinite family of graphs is also provided here to demonstrate that, typically, the spread and the sub-Gaussian constants differ by an order of magnitude.