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SUMMARY:Quantum Chaos\, Random Matrices\, and Spread Complexity of Time Ev
 olution - Vijay Balasubramanian
DTSTART:20250602T140000Z
DTEND:20250602T150000Z
UID:TALK231385@talks.cam.ac.uk
CONTACT:122734
DESCRIPTION:I will describe a measure of quantum state complexity defined 
 by minimizing the spread of the wavefunction over all choices of basis. We
  can efficiently compute this measure\, which displays universal behavior 
 for diverse chaotic systems including spin chains\, the SYK model\, and qu
 antum billiards.  In the minimizing basis\, the Hamiltonian is tridiagonal
 \, thus representing the dynamics as if they unfold on a one-dimensional c
 hain. The recurrent and hopping matrix elements of this chain comprise the
  Lanczos coefficients\, which I will relate through an integral formula to
  the density of states. For Random Matrix Theories (RMTs)\, which are beli
 eved to describe the energy level statistics of chaotic systems\, I will a
 lso derive an integral formula for the covariances of the Lanczos coeffici
 ents. These results lead to a conjecture: quantum chaotic systems have Lan
 czos coefficients whose local means and covariances are described by RMTs.
  I will apply this formalism to the Double Scaled SYK (DSSYK) model with i
 s dual to JT gravity and explain that: (a) wormhole length in JT gravity i
 s dual to spread complexity in the DSSYK theory\, (b) the spread complexit
 y increases linearly at early times matching the classical JT geometry\, (
 c) the spread complexity saturates at late time implying that the classica
 l description of the bulk theory fails because the wavefunction has become
  delocalized in the configuration space of gravity.
LOCATION:MR12
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