The problem asks for a rigorous, quantitatively sharp lower bound on the critical aspect ratio k* (width-to-length ratio) of hard rectangles on the integer lattice Z^2 above which a nematic phase provably exists at sufficiently high density, and to determine whether this threshold converges to the mean-field prediction as the rod length n tends to infinity. Current rigorous results establish existence of nematic order for aspect ratios beyond some computable bound, but this bound is not believed to be sharp, and the gap between rigorous results and the predicted threshold remains substantial. The precise question is: does the critical fugacity at which nematic order emerges, as a function of rod length n, admit a sharp asymptotic of the form c/n for an explicit constant c that can be identified from a mean-field or Bethe-lattice approximation, and can this be proved rigorously?
This problem is open because the techniques used to prove nematic ordering, primarily cluster expansion and Peierls-type contour arguments adapted to anisotropic hard-core systems, break down near the actual transition point. The hard-core constraint creates intricate combinatorial dependencies between rod configurations that resist direct probabilistic decoupling. Establishing sharpness requires control over fluctuations at all scales, not just at low density where expansions converge, and it requires ruling out intermediate phases such as smectic or columnar order that could preempt the nematic transition. The lattice geometry introduces additional complications absent in continuum models, since orientational and translational degrees of freedom are coupled in a discrete and non-trivial way.
Solving this problem would represent a major step toward a complete rigorous phase diagram for lattice hard-core systems, a foundational class of models in statistical mechanics. It would provide a template for proving sharp thresholds in other lattice models exhibiting spontaneous symmetry breaking under excluded-volume interactions, and it would clarify the extent to which mean-field theory is quantitatively reliable for anisotropic lattice gases. The techniques developed would likely have implications for rigorous treatments of liquid crystal models more broadly, including those with continuous orientation variables.
This problem emerges directly from the paper on rigorous bounds for nematic phase formation in hard rod and hard rectangle systems on Z^2, which establishes that nematic order exists for sufficiently large aspect ratios but leaves open the sharpness of the derived threshold. The question of sharp thresholds is also connected thematically to the KAM tori paper, since both concern the precise boundary between ordered and disordered regimes in systems with many interacting components, and the combinatorial structure of the nematic ordering problem has echoes in the algebraic structures studied in the q-Onsager algebra paper, where the interplay between deformation parameters and phase-like transitions in exactly solvable models provides conceptual parallels.