Perfect sampling algorithm for Schur processes

We describe random generation algorithms for a large class of random combinatorial objects called Schur processes, which are sequences of random (integer) partitions subject to certain interlacing conditions. This class contains several fundamental combinatorial objects as special cases, such as plane partitions, tilings of Aztec diamonds, pyramid partitions and more generally steep domino tilings of the plane. Our algorithm, which is of polynomial complexity, is both exact (i.e. the output follows exactly the target probability law, which is either Boltzmann or uniform in our case), and entropy optimal (i.e. it reads a minimal number of random bits as an input). The algorithm encompasses previous growth procedures for special Schur processes related to the primal and dual RSK algorithm, as well as the famous domino shuffling algorithm for domino tilings of the Aztec diamond. It can be easily adapted to deal with symmetric Schur processes and general Schur processes involving infinitely many parameters. It is more concrete and easier to implement than Borodin's algorithm, and it is entropy optimal. At a technical level, it relies on unified bijective proofs of the different types of Cauchy and Littlewood identities for Schur functions, and on an adaptation of Fomin's growth diagram description of the RSK algorithm to that setting. Simulations performed with this algorithm suggest interesting limit shape phenomena for the corresponding tiling models, some of which are new.Comment: 26 pages, 19 figures (v3: final version, corrected a few misprints present in v2

Microstructural enrichment functions based on stochastic Wang tilings

This paper presents an approach to constructing microstructural enrichment functions to local fields in non-periodic heterogeneous materials with applications in Partition of Unity and Hybrid Finite Element schemes. It is based on a concept of aperiodic tilings by the Wang tiles, designed to produce microstructures morphologically similar to original media and enrichment functions that satisfy the underlying governing equations. An appealing feature of this approach is that the enrichment functions are defined only on a small set of square tiles and extended to larger domains by an inexpensive stochastic tiling algorithm in a non-periodic manner. Feasibility of the proposed methodology is demonstrated on constructions of stress enrichment functions for two-dimensional mono-disperse particulate media.Comment: 27 pages, 12 figures; v2: completely re-written after the first revie

Image Sampling with Quasicrystals

We investigate the use of quasicrystals in image sampling. Quasicrystals produce space-filling, non-periodic point sets that are uniformly discrete and relatively dense, thereby ensuring the sample sites are evenly spread out throughout the sampled image. Their self-similar structure can be attractive for creating sampling patterns endowed with a decorative symmetry. We present a brief general overview of the algebraic theory of cut-and-project quasicrystals based on the geometry of the golden ratio. To assess the practical utility of quasicrystal sampling, we evaluate the visual effects of a variety of non-adaptive image sampling strategies on photorealistic image reconstruction and non-photorealistic image rendering used in multiresolution image representations. For computer visualization of point sets used in image sampling, we introduce a mosaic rendering technique.Comment: For a full resolution version of this paper, along with supplementary materials, please visit at http://www.Eyemaginary.com/Portfolio/Publications.htm

Constructions and Noise Threshold of Hyperbolic Surface Codes

We show how to obtain concrete constructions of homological quantum codes based on tilings of 2D surfaces with constant negative curvature (hyperbolic surfaces). This construction results in two-dimensional quantum codes whose tradeoff of encoding rate versus protection is more favorable than for the surface code. These surface codes would require variable length connections between qubits, as determined by the hyperbolic geometry. We provide numerical estimates of the value of the noise threshold and logical error probability of these codes against independent X or Z noise, assuming noise-free error correction