Characterize the singular part of the limit distribution in the loss-of-absolute-continuity regime (c < 0) for Brownian motion on U_N^+

Determine the precise structure of the component singular with respect to the Haar state in the large-N limit of the Brownian motion on the unitary quantum group U_N^+ at times t_c = N ln(√2 N) + c N with c < 0. Identify its support and decomposition so as to fully compute the total-variation cutoff profile in this regime, where the central algebra O(U_N^+)_0 is noncommutative and the singular part does not reduce to a single atom.

Background

The paper constructs a Brownian motion on the free unitary quantum group U_N^+ and computes its cutoff profile in total variation. For times t_c = N ln(√2 N) + c N with c ≥ 0, the authors identify the limit profile via the free Meixner law η_c and the semicircular distribution ν_SC. However, for c < 0, absolute continuity with respect to Haar fails, and the process exhibits a singular component.

In previously studied quantum groups (e.g., O_N^+ and S_N^+), the singular part in the loss-of-absolute-continuity regime was an atom, allowing a full description of the profile. Here, the central algebra is noncommutative and the singular component is more intricate. Because the authors cannot identify it, they only obtain a lower bound for the profile in this regime. Clarifying the singular part is necessary to complete the profile for c < 0.