We report the spectral imaging in the UV to visible range with nanometer scale resolution of closely packed GaN/AlN quantum disks in individual nanowires using an improved custom-made cathodoluminescence system. We demonstrate the possibility to measure full spectral features of individual quantum emitters as small as 1 nm and separated from each other by only a few nanometers and the ability to correlate their optical properties to their size, measured with atomic resolution. The direct correlation between the quantum disk size and emission wavelength provides evidence of the quantum confined Stark effect leading to an emission below the bulk GaN band gap for disks thicker than 2.6 nm. With the help of simulations, we show that the internal electric field in the studied quantum disks is smaller than what is expected in the quantum well case. We show evidence of a clear dispersion of the emission wavelengths of different quantum disks of identical size but different positions along the wire. This dispersion is systematically correlated to a change of the diameter of the AlN shell coating the wire and is thus attributed to the related strain variations along the wire. The present work opens the way both to fundamental studies of quantum confinement in closely packed quantum emitters and to characterizations of optoelectronic devices presenting carrier localization on the nanometer scale.

We introduce the framework of discrete holomorphic functions on t-embeddings of weighted bipartite planar graphs; t-embeddings also appeared under the name Coulomb gauges in a recent work [25]. We argue that this framework is particularly relevant for the analysis of scaling limits of the height fluctuations in the corresponding dimer models. In particular, it unifies both Kenyon's interpretation of dimer observables as derivatives of harmonic functions on T-graphs and the notion of s-holomorphic functions originated in Smirnov's work on the critical Ising model. We develop an a priori regularity theory for such functions and provide a meta-theorem on convergence of the height fluctuations to the Gaussian Free Field.

We show that the winding of the branches in a uniform spanning tree on a planar graph converge in the limit of fine mesh size to a Gaussian free field. The result holds assuming only convergence of simple random walk to Brownian motion and a Russo-Seymour-Welsh type crossing estimate, thereby establishing a strong form of universality. As an application, we prove universality of the fluctuations of the height function associated to the dimer model, in several situations.The proof relies on a connection to imaginary geometry, where the scaling limit of a uniform spanning tree is viewed as a set of flow lines associated to a Gaussian free field. In particular, we obtain an explicit construction of the a.s. unique Gaussian free field coupled to a continuum uniform spanning tree in this way, which is of independent interest.

In this article, we prove that the height function associated with the square-ice model (i.e. the six-vertex model with a = b = c = 1 on the square lattice), or, equivalently, of the uniform random homomorphisms from Z 2 to Z, has logarithmic variance. This establishes a strong form of roughness of this height function.

Abstract. We study the Glauber dynamics on the set of tilings of a finite domain of the plane with lozenges of side 1 L. Under the invariant measure of the process (the uniform measure over all tilings), it is well known [4] that the random height function associated to the tiling converges in probability, in the scaling limit L → ∞, to a non-trivial macroscopic shape minimizing a certain surface tension functional. According to the boundary conditions the macroscopic shape can be either analytic or contain "frozen regions" (Arctic Circle phenomenon [3,11]).It is widely conjectured, on the basis of theoretical considerations [23,10], partial mathematical results [25,1] and numerical simulations for similar models ([5], cf. also the bibliography in [25,10]), that the Glauber dynamics approaches the equilibrium macroscopic shape in a time of order L 2+o(1) . In this work we prove this conjecture, under the assumption that the macroscopic equilibrium shape contains no "frozen region".