Conjecture on Tracy–Widom fluctuations in the i.i.d. model

Let $G_n$ be the hexagonal-lattice rhombus with $2n-1$ levels, and let $H_n$ be its weighted adjacency operator. In the i.i.d. model, all edge weights are independent copies of a nonzero almost surely real-valued random variable $X$ satisfying the stated exponential moment conditions. Let $\bar f$ and $\bar g$ denote appropriate constants for the distribution of $X$. The i.i.d. model conjecture. Theorem's Tracy–Widom limit for the mixed model also holds in the i.i.d. model, with $\bar f$ and $\bar g$ replaced by appropriate constants; equivalently, the centered and scaled smallest positive eigenvalue has the corresponding GUE Tracy–Widom limiting distribution. The mixed-model theorem establishes this fluctuation behavior for Gamma-distributed blue-edge weights and unit red-edge weights, whereas the analogous result for general i.i.d. edge weights remains conjectural.