Spectral statistics and energy-gap scaling in $k-$local spin Hamiltonians

We investigate the spectral properties of all-to-all interacting spin Hamiltonians acting on exactly $k$ spins, whose coupling coefficients are drawn from a normal distribution with mean $\mu$ and variance $\sigma^2$. For $\mu = 0$, we demonstrate that the random matrix ensemble -- Gaussian Orthogonal Ensemble (GOE), Gaussian Unitary Ensemble (GUE), or Gaussian Symplectic Ensemble (GSE) -- is determined by the parity of system size $L$ and locality $k$, following standard time-reversal symmetry classification. For couplings with a nonzero mean, we map the Hamiltonians to deformed random matrix ensembles and analyze conditions for an energy gap between the ground state and the first excited state. For $\mu < 0$, we find two distinct regimes: for $k \gg \sqrt{L}$, the gap closes at critical disorder $\sigma_{c} \approx |\mu|$. Near this transition the energy gap $\Delta$ exhibits universal quadratic scaling $\Delta /L \sim (\sigma - \sigma_{c})^{2}$. When $k \ll \sqrt{L}$, $\sigma_{c}$ scales with $|\mu|$, but lacks a sharp transition. Our work introduces a semi-solvable model that captures universal features of random-matrix statistics, and spectral gap formation, providing a foundation for systematic extensions to more general many-body systems.