Diagonal monitored-transport limit

Let $B_{L_N}=(PS_{L_N})(PS_{L_N-1})\cdots(PS_1)$, where $S_1,\ldots,S_{L_N}$ are independent Haar-distributed unitary matrices in $U(N)$ and $P$ is a deterministic orthogonal projection of rank $N-1$. Let $\mu_\tau$ be the compactly supported probability measure on $[0,1]$ characterized by

$$S_{\mu_\tau}(z)=\exp\left(\frac{\tau}{1+z}\right).$$

Diagonal monitored-transport limit. If $L_N/N\to\tau$, then

$$\operatorname{ESD}(B_{L_N}^\dagger B_{L_N})\xrightarrow{\mathrm{prob}}\mu_\tau.$$

The fixed-length random-matrix limit and the free small-loss limit support this prediction, but the regime in which the chain length and matrix dimension grow simultaneously is described in the source as conjectural and is not established by those results.