Ergodicity and degenerate limit shape for multiplicative statistics with rad_1<1

Let $rad_1<1$ in the setting of Theorem~, and suppose that $f$ has a pole at $rad_1$. Let $mu^{(n)}$ denote the corresponding measures and let $\alpha^{(n)}$ be their scaling function. Degenerate-limit-shape conjecture. The measures $mu^{(n)}$ should be ergodic with the choice

$$\alpha^{(n)}\equiv1,$$

which should lead to the degenerate limit shape

$$\varphi(t)=\mathbf{1}(t\in[0,1]).$$

The preceding theorem establishes ergodicity when $rad_1>1$, or when $rad_1=1$ and $f$ has an isolated pole at $1$; for $rad_1<1$, the grand canonical ensemble is nonergodic, while the corresponding canonical measures are suggested to remain ergodic with this degenerate limit shape.