LDP for centered empirical measure of fixed-trace GUE eigenvalues

Prove that the centered empirical measure CL_n of the eigenvalues of the fixed-trace Gaussian Unitary Ensemble (GUE), scaled by 1/√n, satisfies a large deviation principle in (P_1(R),W_1) with good rate function I(μ)=I_log(μ)+½(1−m_2(μ)) if m_1(μ)=0 and m_2(μ)≤1, and I(μ)=∞ otherwise, where I_log(μ)=−∬ log|x−y| μ(dx)μ(dy)+½∫ x^2 μ(dx)−3/4 and m_k(μ)=∫ x^k μ(dx).

Background

The authors strengthen the Biane–Voiculescu transport inequality on the Carlen–Gangbo sphere and draw parallels with large deviations for Wigner matrices.

Motivated by the strengthened inequality, they conjecture an LDP for the centered empirical measure of eigenvalues under the fixed-trace GUE ensemble, analogous to the classical LDP with I_log for standard GUE, but with a constraint-adjusted rate function capturing centering and energy constraints.

References

We conjecture that $\mathsf{C}L_n$ satisfies a large deviation principle in $(P_1(\mathbb{R}),W_1)$ with good rate function $\mu \mapsto \begin{cases} I_{\mathrm{log}}(\mu) + \tfrac12\big(1 - m_2(\mu)\big) &\text{if } m_1(\mu)=0, \ m_2(\mu)\le 1, \ \infty &\text{otherwise}, \end{cases}$

— Geodesic convexity and strengthened functional inequalities in submanifolds of Wasserstein space (2508.13698 - Chaintron et al., 19 Aug 2025) in Section 5.2 (Biane–Voiculescu inequality)