Kai Hippi

Quantum Mixing and Benjamini–Schramm Convergence of Hyperbolic Surfaces

Quantum chaos studies how quantum systems reflect the chaotic behaviour of their classical counterparts. Two central notions in this field are quantum ergodicity and quantum mixing, which describe how quantum states spread out and how correlations between states behave in the high-energy limit.

Roughly speaking, a system is quantum ergodic if the expectation value of a fixed observable in most high-energy eigenfunctions approaches the classical average. Similarly, a system is quantum mixing if it is quantum ergodic and the associated transition amplitudes between states effectively decay in the high-energy limit. For compact hyperbolic surfaces, quantum ergodicity was proven by Snirelman, Zelditch, and Colin de Verdière. Building on this, Zelditch proved that these surfaces are also quantum mixing.

In analytic number theory, quantum mixing is connected to bounding triple product L-functions arising from the theory of automorphic forms. In random matrix theory, a strong version of quantum mixing was recently proven for large random Wigner matrices as a form of the Eigenstate Thermalisation Hypothesis.

In this talk, I will present a new large-scale analogue of Zelditch's quantum mixing theorem, which complements the large-scale quantum ergodicity theorem of Le Masson and Sahlsten. Instead of studying eigenfunctions on a fixed surface, we fix an energy window and study eigenfunctions of a sequence of growing surfaces with eigenvalues in this window. First, I will present the background leading to these new large-scale quantum mixing results, and then I will present the main ideas of the proof. The talk is based on my recent work available on arXiv (arXiv:2512.15504).