 Modified logarithmic Sobolev inequality

ID: modified-logarithmic-sobolev-inequality

Codex  0 Created 2026-09-24 Updated 2026-09-28

A modified logarithmic Sobolev inequality bounds the entropy of

e^{λ Z}

by a one-sided coordinate-difference energy. For independent coordinates and arbitrary coordinate-deleted functions

f_{i}

, tensorization gives

Ent (e^{λ f}) \leq E [e^{λ f} \sum_{i} ϕ (- λ (f - f_{i}))], ϕ (u) = e^{u} - u - 1.

If that energy is at most

v

, one common normalization is

Ent (e^{λ Z}) \leq \frac{λ ^{2} v}{2} E e^{λ Z} .

 New to topics? Read the docs here!