Thermodynamic Quantum Chaos and large networks

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Thermodynamic Quantum Chaos and large networks

Group Analysis and Dynamical Systems

Quantum chaos is a field that is aimed to study the properties of eigenfunctions of the Laplacian (stationary quantum states) on a Riemannian manifold using the chaotic properties of the underlying geodesic flow in high energy, that is, in the large eigenvalue limit. In this sense the field is connecting quantum mechanics to classical mechanics. A key result in the field is the Quantum Ergodicity Theorem of Shnirelman, Zelditch and Colin de Verdière, which is an equidistribution result of the eigenfunctions for large eigenvalues when the geodesic flow is ergodic. In our recent work we have been attempting to study the theory for a problem of Thermodynamic Quantum Ergodicity (TQE), where instead of large energy, we fix an energy window and vary the geometric properties of the manifold such as volume or genus. The project would aim to develop TQE, in particular for the context of Lie groups and variable curvature manifolds. Moreover, we would attempt to find connect the ideas from TQE to discrete analogues such as spectral theory of large networks, which are well-developed by the recent works of Anantharaman, Brooks, Le Masson, Lindenstrauss, Sabri, and others.

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