We prove the existence of infinitely many low-lying and fundamental closed geodesics on the modular surface which are reciprocal, that is, invariant under time reversal. The method combines ideas from parts I and II of this series, namely, the dispersion method in bilinear forms, as applied to thin semigroups coming from restricted continued fractions.

In this article we use cluster structures and mirror symmetry to explicitly describe a natural class of Newton–Okounkov bodies for Grassmannians. We consider the Grassmannian $\mathbb{X}={\mathit{Gr}}_{n-k}({\mathbb{C}}^n)$, as well as the mirror dual Landau–Ginzburg model $({\check{\mathbb{X}}}^{\circ },W:{\check{\mathbb{X}}}^{\circ }\to \mathbb{C})$, where ${\check{\mathbb{X}}}^{\circ }$ is the complement of a particular anticanonical divisor in a Langlands dual Grassmannian $\check{\mathbb{X}}={\mathit{Gr}}_k(({\mathbb{C}}^n{)}^{\ast })$ and the superpotential $W$ has a simple expression in terms of Plücker coordinates. Grassmannians simultaneously have the structure of an $\mathcal{A}$-cluster variety and an $\mathcal{X}$-cluster variety; roughly speaking, a cluster variety is obtained by gluing together a collection of tori along birational maps. Given a plabic graph or, more generally, a cluster seed $G$, we consider two associated coordinate systems: a network or $\mathcal{X}$-cluster chart ${\Phi }_G:({\mathbb{C}}^{\ast }{)}^{k(n-k)}\to {\mathbb{X}}^{\circ }$ and a Plücker cluster or $\mathcal{A}$-cluster chart ${\Phi }_G^{\vee }:({\mathbb{C}}^{\ast }{)}^{k(n-k)}\to {\check{\mathbb{X}}}^{\circ }$. Here ${\mathbb{X}}^{\circ }$ and ${\check{\mathbb{X}}}^{\circ }$ are the open positroid varieties in $\mathbb{X}$ and $\check{\mathbb{X}}$, respectively. To each $\mathcal{X}$-cluster chart ${\Phi }_G$ and ample boundary divisor $D$ in $\mathbb{X}\setminus {\mathbb{X}}^{\circ }$, we associate a Newton–Okounkov body ${\Delta }_G(D)$ in ${\mathbb{R}}^{k(n-k)}$, which is defined as the convex hull of rational points; these points are obtained from the multidegrees of leading terms of the Laurent polynomials ${\Phi }_G^{\ast }(f)$ for $f$ on $\mathbb{X}$ with poles bounded by some multiple of $D$. On the other hand, using the $\mathcal{A}$-cluster chart ${\Phi }_G^{\vee }$ on the mirror side, we obtain a set of rational polytopes—described in terms of inequalities—by writing the superpotential $W$ as a Laurent polynomial in the $\mathcal{A}$-cluster coordinates and then tropicalizing. Our first main result is that the Newton–Okounkov bodies ${\Delta }_G(D)$ and the polytopes obtained by tropicalization on the mirror side coincide. As an application, we construct degenerations of the Grassmannian to normal toric varieties corresponding to (dilates of ) these Newton–Okounkov bodies. Our second main result is an explicit combinatorial formula in terms of Young diagrams, for the lattice points of the Newton–Okounkov bodies, in the case in which the cluster seed $G$ corresponds to a plabic graph. This formula has an interpretation in terms of the quantum Schubert calculus of Grassmannians.

We consider the probability of having two intervals (gaps) without eigenvalues in the bulk scaling limit of the Gaussian unitary ensemble of random matrices. We describe uniform asymptotics for the transition between a single large gap and two large gaps. For the initial stage of the transition, we explicitly determine all the asymptotic terms (up to the decreasing ones) of the logarithm of the probability. We obtain our results by analyzing double-scaling asymptotics of a Toeplitz determinant whose symbol is supported on two arcs of the unit circle.