Tracy–Widom fluctuations of the frozen boundary in large ASMs

Establish that the fluctuations of the frozen boundary of uniformly sampled large n×n alternating sign matrices are governed by the GUE Tracy–Widom distribution; equivalently, prove that the properly centered and scaled deviations of the frozen boundary from the arctic curve converge in distribution to the Tracy–Widom F2 law as n → ∞.

Background

The authors discuss the limit shape (arctic curve) for ASMs and analyze fluctuations via a conjectural Fredholm determinant representation for B_{n,s}. Under this conjecture, they derive convergence to the GUE Tracy–Widom distribution near the diagonal intersection point of the arctic curve.

They emphasize that a rigorous derivation of their main conjecture would close a gap in proving Tracy–Widom fluctuations for ASMs, identifying this as a long-standing open question.