Modified logarithmic Sobolev inequality

A modified logarithmic Sobolev inequality bounds the entropy of $e^{\lambda Z}$ by a one-sided coordinate-difference energy. For independent coordinates and arbitrary coordinate-deleted functions $f_i$, tensorization givesIf that energy is at most $v$, one common normalization is

The Herbst argument integrates a logarithmic Sobolev bound on $\lambda H^{\prime }(\lambda )-H(\lambda )$ to control the centered cumulant-generating function $H$, then applies a Chernoff bound to obtain concentration.