Applicability of Biane’s recovery method in the non-atomic singular case for U_N^+

Ascertain whether Biane’s method for recovering the absolutely continuous part of a measure from its moments, when the singular part is a single atom, can be applied to 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, where the singular part is not a single atom.

Background

The paper reviews that in earlier work the singular part in the loss-of-absolute-continuity regime often reduces to a single atom, enabling the use of Biane’s technique to recover the absolutely continuous component from moments. In the present setting for U_N^+, the singular part is more complicated than a sole atom.

Because of this complexity, the authors cannot directly apply Biane’s approach and explicitly state uncertainty about its applicability. Establishing whether or how this method can be adapted to the non-atomic singular part would provide a pathway to a complete description of the limit profile for c < 0.