Talagrand’s Inequality in Hereditary Settings

We develop a nicely packaged form of Talagrand’s inequality that can be applied to prove concentration of measure for functions defined by hereditary properties. We illustrate the framework with several applications from combinatorics and algorithms. We also give an extension of the inequality valid in spaces satisfying a certain negative dependence property and give some applications. 1 Talagrand’s Inequality Talagrand’s inequality is an isoperimetric inequality that applies in the setting where Ω = ∏ i∈I Ωi is a product space indexed by some finite index set I with the product measure. ∗Submitted to Random Structures and Algorithms and under revision. †Work done while at the SPIC Mathematical Institute, Chennai, India and while visiting BRICS, Basic Research in Computer Science, Centre of the Danish National Research Foundation, Department of Computer Science, University of Aarhus, Denmark.