Dimension Monotonicity in Laguerre Ensembles I: Fractional Moments and Shape Transitions in the Unitary Case
Abstract
Let WN,N+λ have the Laguerre unitary distribution with size N and real shape λ≥0. For s>0, we consider the normalized moment Cs,λ(N)=N−s−1𝔼[TrWN,N+λs] and its dimension decrement ΓN,s,λ=Cs,λ(N)−Cs,λ(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: Cs,0(N) decreases for 0<s<1 and s>2, increases for 1<s<2, and is constant for s∈{1,2}. For every s>0 and every N, the decrement is strictly increasing in λ. 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 NλN is of order one, with critical constant τs*=(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.
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BibTeX
@misc{hutnik2026dimension,
title = {{Dimension Monotonicity in Laguerre Ensembles I: Fractional Moments and Shape Transitions in the Unitary Case}},
author = {Ondrej Hutník},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.12147},
eprint = {2608.12147},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.12147},
}
RIS
TY - UNPB
TI - Dimension Monotonicity in Laguerre Ensembles I: Fractional Moments and Shape Transitions in the Unitary Case
AU - Ondrej Hutník
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2608.12147
UR - https://arxiv.org/abs/2608.12147
AB - Let W_N,N+λ have the Laguerre unitary distribution with size N and real shape λ≥0. For s>0, we consider the normalized moment C_s,λ(N)=N^−s−1𝔼[TrW_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 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∈{1,2}. For every s>0 and every N, the decrement is strictly increasing in λ. 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 Nλ_N is of order one, with critical constant τ_s^*=(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.
ER -