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Quantum Unique Ergodicity for Cayley Graphs of Quasirandom Groups
Michael Magee1, Joe Thomas1, Yufei Zhao2
1Department of Mathematical Sciences, Durham University, Lower Mountjoy, DH1 3LE Durham, UK.
Communications in Mathematical Physics
|August 22, 2023
Summary
Highly quasirandom finite groups, where irreducible representations are large, ensure Cayley graphs distribute eigenfunction mass effectively. This property aids in understanding group structure through spectral analysis.
Area of Science:
- Group Theory
- Representation Theory
- Spectral Graph Theory
Background:
- Definition of C-quasirandom finite groups: irreducible complex representations have dimension at least C.
- Introduction of quantum probability measure associated with unit functions on finite groups.
- Relevance of group properties to the spectral analysis of their associated graphs.
Purpose of the Study:
- To investigate the spectral properties of Cayley graphs for highly quasirandom finite groups.
- To demonstrate how the quasirandomness of a group influences the distribution of mass for eigenfunctions.
- To establish a connection between group-theoretic properties and graph-theoretic spectral measures.
Main Methods:
- Utilizing the definition of C-quasirandom groups.
- Associating quantum probability measures with eigenfunctions of the adjacency operator.
- Analyzing the mass distribution of these measures on specific subsets of the group.
Main Results:
- For highly quasirandom groups, Cayley graphs possess orthonormal eigenbases for their adjacency operators.
- The quantum probability measures of eigenfunctions concentrate a significant proportion of their mass on selected subsets.
- This mass concentration is shown to be close to the 'correct' proportion, indicating structured behavior.
Conclusions:
- The quasirandomness of a finite group has a direct impact on the spectral properties of its Cayley graphs.
- Eigenfunctions of Cayley graphs of quasirandom groups exhibit controlled mass distribution, linking spectral and group-theoretic properties.
- This work provides a framework for understanding graph properties through the lens of group representation theory.
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