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

Abstract

Let $W_{N,N+\lambda}$ have the Laguerre unitary distribution with size $N$ and real shape $\lambda\ge0$. For $s>0$, we consider the normalized moment $$C_{s,\lambda}(N)=N^{-s-1}\mathbb{E}[\operatorname{Tr}W_{N,N+\lambda}^{s}]$$ and its dimension decrement $\Gamma_{N,s,\lambda}=C_{s,\lambda}(N)-C_{s,\lambda}(N+1)$. Iterating the Laguerre moment recurrence separates this decrement into a square source and a nonnegative shape source. The square source gives the complete finite-dimensional sign diagram: $C_{s,0}(N)$ decreases for $0<s<1$ and $s>2$, increases for $1<s<2$, and is constant for $s\in\{1,2\}$. For every $s>0$ and every $N$, the decrement is strictly increasing in $\lambda$. The same decomposition determines the critical shrinking-shape scales: $N^{-2s}$ for $0<s<1/2$, $(\log N)/N$ for $s=1/2$, and $N^{-1}$ for $s>1/2$. In the convex range $1<s<2$, where the two sources have opposite signs, the transition occurs when $N\lambda_N$ is of order one, with critical constant $$\tau_s^*=\frac{s(s-1)(2-s)}{6(2s-1)}.$$ In the convex range, both crossings are unique for every finite $N$, and we determine their locations to second order. At $s=1/2$ we also obtain a bounded-shape two-term expansion, which supplies the unitary estimates used in the companion orthogonal paper.