Throughout this thesis, we are going to work in a hard disk model which is analogue to the setting in [MR09] and [HMR13]. In this model of two-dimensional crystals, every admissible configuration is a hard disk configuration, which, in addition, is a perturbed version of some triangular lattice with side length one. A triangular lattice with side length one is called standard. The objective of ...

Covering a graph or a lattice with nonoverlapping dimers is a problem that has received considerable interest in areas, such as discrete mathematics, statistical physics, chemistry, and materials science. Yet, the problem of percolation on dimer-covered lattices has received little attention. In particular, percolation on lattices that are fully covered by nonoverlapping dimers has not evidentl...

The selection of the ground state among nearly degenerate states due to quantum fluctuations is studied for the S = 1/2 XY-like Heisenberg antiferromagnets on the triangular lattice in the magnetic field applied along the hard axis, which was first pointed out by Nikuni and Shiba. We find that the selected ground state sensitively depends on the degree of the anisotropy and the magnitude of the...

The Hofstadter problem is studied on the hexagonal lattice. We first establish a relation between the spectra for the hexagonal lattice and for its dual lattice, the triangular lattice. Following the idea of Faddeev and Kashaev, we then obtain the Bethe-Ansatz equations for this system.

It is well-known that plane partitions, lozenge tilings of a hexagon, perfect matchings on a honeycomb graph, and families of non-intersecting lattice paths in a hexagon are all in bijection. In this work we consider regions that are more general than hexagons. They are obtained by further removing upward-pointing triangles. We call the resulting shapes triangular regions. We establish signed v...

Formation of random network of DNA strands is simulated on a two-dimensional triangular lattice. We investigate the size distribution of pores in the network. The results are interpreted within theory of perco-lation on Bethe lattice.

This paper presents a general inequality for the grating lobes of the planar phased array, whether rectangular lattice or triangular lattice. And for the planar phased array with grating lobes, the maximum scanning angle is given.

Recently Beaton, de Gier and Guttmann proved a conjecture of Batchelor and Yung that the critical fugacity of self-avoiding walks interacting with (alternate) sites on the surface of the honeycomb lattice is 1 + √ 2. A key identity used in that proof depends on the existence of a parafermionic observable for self-avoiding walks interacting with a surface on the honeycomb lattice. Despite the ab...