Establish uniform integrability for the fixed-filling character ensemble

Establish uniform integrability of the $2\alpha$-th character moments for the discrete fixed-filling $C_N$ ensemble in the unlocked phase, so that the proposed Gaussian character central-limit law rigorously implies the thermodynamic ratio $\mathcal R_\alpha^{(\eta)}$.

Background

The opposite-fixed/equal-fixed normalization-free ratio is represented exactly by a $2\alpha$-th moment of the fundamental symplectic character in a discrete $C_N$ ensemble. The paper proposes that, for $0<\alpha<4$, this character converges to a centered Gaussian with variance $1/\alpha$.

Weak convergence alone does not ensure convergence of the relevant absolute moments. Uniform tail bounds, or an equivalent uniform-integrability result, are required to exchange the thermodynamic limit with the $2\alpha$-th moment. The authors state that this condition is currently unavailable for the fixed-filling grid.