2025/07/11 by Tim Austin, Austin, Tim · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Neural Networks and Applications #Operator Algebras (math.OA) #Probability (math.PR) #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2507.08909
openalex publication_date 2025/07/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This work studies certain notions of entropy that can be associated to (i) a representation of a separable, unital C*-algebra \mathfrakA and (ii) an auxiliary random sequence (πn)n≥ 1 of finite-dimensional representations of \mathfrakA. This continues a previous research program into the properties of these entropy notions when each πn 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 πn at all. After developing some of this general theory, we then focus on the special case in which \mathfrakA is the group C*-algebra of a finitely-generated free group and each πn 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.