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 $\lambda \ge 0$. For $s>0$, we consider the normalized moment $$C_{s,\lambda }(N)=N^{-s-1}\mathbb{E}[\mathrm{Tr}{W}_{N,N+\lambda }^s]$$ 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 $${\tau }_s^{\ast }=\frac{s(s-1)(2-s)}{6(2s-1)}.$$ 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.

v2: Substantially revised and expanded version with a new title. The original half-moment result from v1 is retained, but the proof is replaced by a recurrence-based two-parameter theory covering general positive moments, strict shape monotonicity, critical shrinking-shape scales, and the convex transition. v1: Original proof in the complex setting, based on Abreu's recurrence