2026/07/21 by Yukun He
#math-ph #hep-th #math.MP #math.OA #math.PR
We consider the Sachdev--Ye--Kitaev model of N Majorana fermions with random q-body interactions. For q=4, we show that as N→ ∞ through even integers, the largest eigenvalue of the model satisfies (λ1)/(√(N))→ 4∫0^∞ g0(t)4 dt ≈ 0.32504 almost surely , where g0(t)=(1)/(2)∫ e-Etρ0(dE) is the unique solution of the zero-temperature quartic Schwinger--Dyson equation for which ρ0 is a probability measure, ∫ E2ρ0(dE)=1/4, and g03∈ L1(0,∞). The main ingredient of the proof is the calculation of the SYK free-energy limit at every fixed positive temperature. We achieve this by introducing a new finite-bath interpolation, which reduces the quartic SYK pressure problem to a local cavity-kernel identity valid at every such temperature. We then identify the zero-temperature slope of the Schwinger--Dyson pressure and transfer it to the spectral edge. GPT-5.6 assisted with literature search, the development of technical arguments, and manuscript preparation; the author is responsible for the contents.