Spectral-norm asymptotics of the SYK Hamiltonian
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What is the leading expected spectral norm of the SYK Hamiltonian when the even interaction order k grows to infinity but k=o(√n)? The proposed scale is √(2n)/k.
Why it matters and what counts as progress
Why it matters: Matching the spectral edge closes a factor-k gap between previous bounds in this growing-interaction regime.
What counts as progress: Matching leading upper and lower estimates under the specified SYK normalization; a sparse extension must retain its additional assumptions.
How the problem developed
- 30 Jan 2018 · Partial result [1]
Renjie Feng, Gang Tian and Dongyi Wei Proved convergence of the global eigenvalue density for the SYK model and studied its largest eigenvalue at fixed interaction order.
- 29 Jul 2026 · Asymptotic bound [2]
Anindya Basu, Pravesh K. Kothari and Shyam Narayanan Midha Proved E‖H_SYK‖=(1−o(1))√(2n)/k for superconstant even interaction order k=o(√n), using a bosonic witness and spectral comparison.
AI contribution
Sharp Bounds on Ground State Energy of the SYK Model
GPT-5.6 Pro suggested a twisted-boson moment identity and an isometric embedding and intertwining identity that transfer a bosonic witness into a lower-bound certificate. These steps support the authors’ E||H_SYK||=(1−o(1))√(2n)/k result for even k growing to infinity with k=o(√n), with a stated sparse extension.
Sources
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