Dimension Monotonicity in Laguerre Ensembles I: Fractional Moments and Shape Transitions in the Unitary Case

Let $W_{N,N+λ}$ have the Laguerre unitary distribution with size $N$ and real shape $λ\ge0$. For $s>0$, we consider the normalized moment $$C_{s,λ}(N)=N^{-s-1}\mathbb{E}[\operatorname{Tr}W_{N,N+λ}^{s}]$$ and its dimension decrement $Γ_{N,s,λ}=C_{s,λ}(N)-C_{s,λ}(N+1)$. Iterating the Laguerre moment r