A recent study has shown that the normalized generalized Brézin–Gross–Witten (BGW) correlators, which arise in a matrix model of two‑dimensional quantum gravity, always curve downward when the genus and the total degree of the interaction are held constant. The authors used a formula for connected $n$‑point functions of tau‑functions of the BKP hierarchy, supplied by Wang and the first author of the paper, to prove that the second difference of the generalized BGW coefficients is negative. This mathematical property guarantees strict concavity.

The concavity result led the authors to establish strict Schur‑concavity, a stronger form of symmetry. They also derived sharp upper and lower bounds for the correlators, showing that for any integer $s≥1$ and any positive partition $d$ of size $s$, the normalized generalized BGW correlator satisfies $$C_h(s)\, ext{≤}\,C_h(d)\, ext{≤}\,C_h(1,s) \\text{for all}\,0≤h≤s+1.$$ The inequalities hold regardless of the specific shape of the partition, because the relevant generalized BGW coefficients are always ordered by the dominance order of partitions.

The paper does not mention any explicit limitations on the applicability of the concavity result; the theorem is stated for all fixed genera and total degrees covered by the generalized BGW model.

These findings resolve a conjecture about the monotonicity of ordinary BGW correlators and give a new mathematical foothold for studying phase transitions in related random‑matrix models.