Non-commutative probability insights into the double-scaling limit SYK Model with constant perturbations: moments, cumulants, and $q$-independence

Extending the results of \cite{Wu}, we study the double-scaling limit SYK (DSSYK) model with an additional diagonal matrix with a fixed number $c$ of nonzero constant entries $\theta$. This constant diagonal term can be rewritten in terms of Majorana fermion products. Its specific formula depends on the value of $c$. We find exact expressions for the moments of this model. More importantly, by proposing a moment-cumulant relation, we reinterpret the effect of introducing a constant term in the context of non-commutative probability theory. This gives rise to a $\tilde{q}$ dependent mixture of independences within the moment formula. The parameter $\tilde{q}$, derived from the q-Ornstein-Uhlenbeck (q-OU) process, controls this transformation. It interpolates between classical independence ($\tilde{q}=1$) and Boolean independence ($\tilde{q}=0$). The underlying combinatorial structures of this model provide the non-commutative probability connections. Additionally, we explore the potential relation between these connections and their gravitational path integral counterparts.