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Quenched properties of the spectral form factor
Dimitrios Charamis1,2, Manas Kulkarni3, Jorge Kurchan2
1Université Paris Saclay, CEA, CNRS, IPhT, 91191 Gif-sur-Yvette, France.
Physical Review. E
|February 20, 2026
Summary
The quenched Spectral Form Factor (SFF) is not self-averaging, unlike its annealed counterpart. Its average and fluctuations align with Gumbel distributions, even for non-Hermitian systems.
Area of Science:
- Quantum Chaos
- Statistical Mechanics
- Random Matrix Theory
Background:
- The Spectral Form Factor (SFF) characterizes spectral statistics in quantum systems.
- Generalizations of SFF exist for both Hermitian and non-Hermitian matrices.
- SFF is known to not be self-averaging.
Purpose of the Study:
- Investigate the properties of the quenched SFF, specifically its logarithm and average.
- Compare the quenched SFF with its annealed counterpart.
- Analyze the fluctuations of the quenched SFF at late times.
Main Methods:
- Studied the average and logarithm of the quenched SFF.
- Compared quenched and annealed averages of SFF and its logarithm.
- Analyzed SFF fluctuations using a variable transformation compatible with Gumbel distribution.
- Examined Fisher zeros of the partition function.
Main Results:
- The quenched SFF is self-averaging for both Hermitian and non-Hermitian cases.
- Quenched and annealed averages of SFF coincide up to subleading constants at high temperatures.
- Fluctuations of the quenched SFF logarithm are deep, exhibiting spikes near partition function zeros.
- A variable transformation of the quenched SFF logarithm is compatible with a Gumbel distribution.
Conclusions:
- The exponential tail of the Gumbel distribution arises from sampling deep spikes (Fisher zeros) of the quenched SFF.
- The findings hold for non-Hermitian Hamiltonians and random matrices, extending previous results.
- This study provides insights into spectral statistics in complex quantum systems.
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