Note on the $q=2$ $R$-para-fermionic SYK model

We investigate the $q=2$ SYK model with $R$-para-particles ($R$-PSYK$_2$), analyzing its thermodynamics and spectral form factor (SFF) using random matrix theory. The Hamiltonian is quadratic, with coupling coefficients randomly drawn from the Gaussian Unitary Ensemble (GUE). The model displays self-averaging behavior and exhibits an exponential ramp in its SFF dynamics: $\mathcal{K}(t) \sim e^{C_0t}$. The growth rate $C_0$ tends toward either a constant or infinity in the $N\to \infty$ limit, depending on specific statistics of the model. These results provide novel perspectives on quantum systems with unconventional statistics.