Conjecture on variance and ergodic capacity in canonical and Kronecker channels

Let $V_{ {\sf can} }(\rho)$ and $V_{ {\sf kron} }(\rho)$ denote the variances of the canonical and Kronecker channel capacities, respectively, and let $C_{{\sf{erg}},\hspace{0.03in} {\sf can} }(\rho)$ and $C_{{\sf{erg}},\hspace{0.03in} {\sf kron} }(\rho)$ denote their corresponding ergodic capacities. Here $N$ is the system size, and $\rho$ is the signal-to-noise ratio.

Variance-smoothing conjecture. For a large class of channels in the medium- to high-$\mathrm{SNR}$ regime,

$$\frac{\sqrt{V_{ {\sf can} }(\rho)}}{C_{{\sf{erg}},\hspace{0.03in} {\sf can} }(\rho)}\stackrel{N\rightarrow\infty}{\longrightarrow}0,\qquad \frac{\sqrt{V_{ {\sf kron} }(\rho)}}{C_{{\sf{erg}},\hspace{0.03in} {\sf kron} }(\rho)}\stackrel{N\rightarrow\infty}{\longrightarrow}0,$$

and

$$V_{ {\sf can} }(\rho)\geq V_{ {\sf kron} }(\rho).$$

The conjecture formalizes the observed smoothing effect of the Kronecker model: capacity fluctuations are asymptotically negligible relative to ergodic capacity, while the canonical channel has at least as much capacity variance as its Kronecker approximation. The source presents this as an open question motivated by numerical studies and leaves its proof for future work.