On measure concentration in graph products

Abstract

Bollobás and Leader [1] showed that among the n-fold products of connected graphs of order k the one with minimal t-boundary is the grid graph. Given any product graph G and a set A of its vertices that contains at least half of V (G), the number of vertices at a distance at least t from A decays (as t grows) at a rate dominated by P(X1 + . . . + Xn \geq t) where Xi are some simple i.i.d. random variables. Bollobás and Leader used the moment generating function to get an exponentialbound for this probability. We insert a missing factor in the estimate by using a somewhat more subtle technique (cf. [3]).

Keywords

• graph product
• discrete isoperimetric inequalities
• concentration of measure
• sums of independent random variables
• tail probabilities
• large deviations