Concentration of measure for finite spin systems

Holger Sambale, Arthur Sinulis

Published 2018-07-20Version 1

In this work we continue the investigation of concentration of measure of higher order for various finite spin systems. We show that under the presence of a logarithmic Sobolev inequality it is possible to estimate the growth of the $L^p$-norms of any function, which leads to concentration inequalities. Applications to several statistics in the exponential random graph models, the random coloring models, the hard-core model and the Erd\"os-Renyi model are given. We show the effect of better concentration results by centering not around the mean of the statistic (a zero order approximation), but around a stochastic term (a first order approximation) in the exponential random graph model. In the Erd\"os-Renyi model we prove a central limit theorem for various subgraph counts.