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Annealed almost periodic entropy

Tim Austin

Abstract

This work studies certain notions of entropy that can be associated to (i) a representation of a separable, unital C*-algebra $\mathfrak{A}$ and (ii) an auxiliary random sequence $(π_n)_{n\ge 1}$ of finite-dimensional representations of $\mathfrak{A}$. This continues a previous research program into the properties of these entropy notions when each $π_n$ is deterministic, which uncovered a range of analogies with entropy in ergodic theory and also with non-commutative generalizations of Szegő's limit theorems. We associate two new notions of entropy to data as in (i) and (ii) above: `annealed' AP entropy, which is roughly a kind of first-moment average of deterministic AP entropies; and `zeroth-order' AP entropy, which controls the large deviations probabilities that certain positive definite functions appear in the representations $π_n$ at all. After developing some of this general theory, we then focus on the special case in which $\mathfrak{A}$ is the group C*-algebra of a finitely-generated free group and each $π_n$ is generated by choosing a tuple of $n$-by-$n$ unitary matrices independently at random from Haar measure. In that case, explicit formulas can be derived for some of our notions of entropy, and new large deviations principles in random matrix theory are obtained as a consequence.

Annealed almost periodic entropy

Abstract

This work studies certain notions of entropy that can be associated to (i) a representation of a separable, unital C*-algebra and (ii) an auxiliary random sequence of finite-dimensional representations of . This continues a previous research program into the properties of these entropy notions when each is deterministic, which uncovered a range of analogies with entropy in ergodic theory and also with non-commutative generalizations of Szegő's limit theorems. We associate two new notions of entropy to data as in (i) and (ii) above: `annealed' AP entropy, which is roughly a kind of first-moment average of deterministic AP entropies; and `zeroth-order' AP entropy, which controls the large deviations probabilities that certain positive definite functions appear in the representations at all. After developing some of this general theory, we then focus on the special case in which is the group C*-algebra of a finitely-generated free group and each is generated by choosing a tuple of -by- unitary matrices independently at random from Haar measure. In that case, explicit formulas can be derived for some of our notions of entropy, and new large deviations principles in random matrix theory are obtained as a consequence.

Paper Structure

This paper contains 70 sections, 102 theorems, 387 equations.

Table of Contents

  1. Introduction
  2. PREPARATIONS
  3. Basic notation and conventions
  4. Linear algebra and linear operators
  5. Real analysis
  6. Large deviations principles
  7. Positive definite matrices and their determinants
  8. Gram matrices and log-determinant entropy
  9. Two-by-two block Gram matrices
  10. Three-by-three-block Gram matrices
  11. Some probability on spaces of matrices
  12. Polar decomposition of integrals
  13. A method of types for Gram matrices
  14. Measure concentration
  15. C*-algebras and their representations
  16. ...and 55 more sections

Key Result

Theorem A

The function $\mathrm{h}^\mathrm{ann}$ takes values in $[-\infty,\infty)$ for each $k$. For any positive definite map $\varphi:\Gamma\to \mathbf{M}_k$, the value $\mathrm{h}^\mathrm{ann}(\varphi)$ is equal to both and (interpreting these as $-\infty$ if any of the determinants appearing here equals $0$).

Theorems & Definitions (216)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Definition 3.1: Large deviations principle
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 206 more