The entropy function of an invariant measure

TL;DR

This paper investigates the growth rate of entropy functions of invariant measures on countable structures, revealing polynomial growth patterns and bounds, and extending previous results in the field.

Contribution

It generalizes known results on entropy growth for invariant measures, provides bounds for infinite languages, and demonstrates arbitrarily fast growth in certain cases.

Findings

Entropy functions grow as $Cn^k + o(n^k)$ for non-redundant measures with finitely many relation symbols.

Existence of invariant measures with entropy functions growing arbitrarily fast in $o(n^k)$ for $k \\ge 2$.

Explicit upper bounds on entropy functions for infinite languages based on relation symbols.

Abstract

Given a countable relational language $L$, we consider probability measures on the space of $L$-structures with underlying set $\mathbb{N}$ that are invariant under the logic action. We study the growth rate of the entropy function of such a measure, defined to be the function sending $n \in \mathbb{N}$ to the entropy of the measure induced by restrictions to $L$-structures on $\{0, \ldots, n-1\}$. When $L$ has finitely many relation symbols, all of arity $k\ge 1$, and the measure has a property called non-redundance, we show that the entropy function is of the form $Cn^k+o(n^k)$, generalizing a result of Aldous and Janson. When $k\ge 2$, we show that there are invariant measures whose entropy functions grow arbitrarily fast in $o(n^k)$, extending a result of Hatami-Norine. For possibly infinite languages $L$, we give an explicit upper bound on the entropy functions of non-redundant invariant measures in terms of the number of relation symbols in $L$ of each arity; this implies that finite-valued entropy functions can grow arbitrarily fast.