Notes on complexity growth rate, grand potential and partition function

• Published in: General relativity and gravitation Vol. 54; no. 5
• Main Authors: Sun, Wei, Ge, Xian-Hui
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-05-2022; Springer Nature B.V
• Subjects: Astronomy; Astrophysics and Cosmology; Black holes; Classical and Quantum Gravitation; Complexity; Differential Geometry; Free energy; Gravity; Growth rate; Mathematical and Computational Physics; Partitions (mathematics); Phase transitions; Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Research Article; Saddle points; Theoretical; Thin walled shells; Grand potential; Partition function; Black holes; SYK model; Complexity
• ISSN: 0001-7701; 1572-9532
• DOI: 10.1007/s10714-022-02933-4

We examine the complexity/volume conjecture and further investigate the possible connections between complexity and partition function. The complexity/partition function relation is then utilized to study the complexity of the thermofield double state of extended SYK models for various conditions.The difference between the complexity/partition function relation with the complexity/action duality comes from the fact that partition function can be evaluated from more than one saddle point, so they differ at most at the non-perturbative level. We further analyze free energy and the growth rate of complexity in the neutral AdS-Vaidya black hole formed by collapsing an uncharged spherically symmetric thin shell of null fluid. The relation between the late-time complexity growth rate and black hole/SYK wormhole phase transition is also discussed. Finally, we check the Lloyd bound with our proposal.