Log-Sobolev Inequality

The log-Sobolev inequality is a functional inequality that strengthens the Poincaré inequality by controlling the entropy of a function rather than merely its variance. Where the Poincaré inequality bounds the L² norm of a function minus its mean by the L² norm of its gradient, the log-Sobolev inequality bounds the relative entropy of a probability distribution by its Dirichlet energy. The result is sub-Gaussian concentration: probabilities decay exponentially rather than polynomially.

The inequality was introduced by Leonard Gross in 1975, who showed that it is equivalent to the hypercontractivity of the associated semigroup. A semigroup satisfies a log-Sobolev inequality if and only if it maps Lᵖ spaces to Lᑫ spaces with q > p for short times — a property with deep implications for the rate of convergence to equilibrium in statistical mechanics and quantum field theory.

In the context of Markov chains, the log-Sobolev constant provides a stronger mixing time bound than the spectral gap alone. A chain with log-Sobolev constant α mixes in time O((1/α) log log n), compared to O((1/λ) log n) for the spectral gap λ. The difference is the transition from polynomial to sub-Gaussian concentration, a qualitative improvement that matters in high-dimensional sampling where the Poincaré bound becomes vacuous.

The log-Sobolev inequality is strictly stronger than the Poincaré inequality: every distribution satisfying a log-Sobolev inequality also satisfies a Poincaré inequality, but the converse fails. The gap between them is the gap between variance control and entropy control, between Gaussian tails and sub-Gaussian tails, between knowing that fluctuations are bounded and knowing that they are exponentially unlikely.

The log-Sobolev inequality is the Poincaré inequality's older sibling: sharper, more demanding, and less universally applicable. But where it applies, it reveals a deeper structure. A system satisfying a log-Sobolev inequality is not merely well-behaved; it is geometrically coherent, with a kind of internal curvature that forces rapid equilibration. The inequality is therefore not just a technical refinement. It is a diagnostic: it distinguishes systems that converge quickly from systems that converge almost instantaneously.