The interaction of a gap with a free boundary in a two dimensional dimer system

Let $\ell$ be a fixed vertical lattice line of the unit triangular lattice in the plane, and let \Cal H be the half plane to the left of $\ell$. We consider lozenge tilings of \Cal H that have a triangular gap of side-length two and in which $\ell$ is a free boundary - i.e., tiles are allowed to protrude out half-way across $\ell$. We prove that the correlation function of this gap near the free boundary has asymptotics $\frac{1}{4\pi r}$, $r\to\infty$, where $r$ is the distance from the gap to the free boundary. This parallels the electrostatic phenomenon by which the field of an electric charge near a conductor can be obtained by the method of images.Comment: 34 pages, AmS-Te

Open boundary Quantum Knizhnik-Zamolodchikov equation and the weighted enumeration of Plane Partitions with symmetries

We propose new conjectures relating sum rules for the polynomial solution of the qKZ equation with open (reflecting) boundaries as a function of the quantum parameter $q$ and the $\tau$-enumeration of Plane Partitions with specific symmetries, with $\tau=-(q+q^{-1})$. We also find a conjectural relation \`a la Razumov-Stroganov between the $\tau\to 0$ limit of the qKZ solution and refined numbers of Totally Symmetric Self Complementary Plane Partitions.Comment: 27 pages, uses lanlmac, epsf and hyperbasics, minor revision

Hopf algebras and Markov chains: Two examples and a theory

The operation of squaring (coproduct followed by product) in a combinatorial Hopf algebra is shown to induce a Markov chain in natural bases. Chains constructed in this way include widely studied methods of card shuffling, a natural "rock-breaking" process, and Markov chains on simplicial complexes. Many of these chains can be explictly diagonalized using the primitive elements of the algebra and the combinatorics of the free Lie algebra. For card shuffling, this gives an explicit description of the eigenvectors. For rock-breaking, an explicit description of the quasi-stationary distribution and sharp rates to absorption follow.Comment: 51 pages, 17 figures. (Typographical errors corrected. Further fixes will only appear on the version on Amy Pang's website, the arXiv version will not be updated.

Towards 48-fold cabling complexity reduction in large flattened butterfly networks

Amongst data center structures, flattened butterfly (FBFly) networks have been shown to outperform their common counterparts such as fat-trees in terms of energy proportionality and cost efficiency. This efficiency is achieved by using less networking equipment (switches, ports, cables) at the expense of increased control plane complexity. In this paper we show that cabling complexity can be further reduced by an order of magnitude, by reconfiguring the optical fully meshed components into optical “pseudo”-fully meshed components. Following established methods, optical star networks are obtained by exchanging the FBFly's regular (grey) optical transceivers for dense wavelength division multiplexing (DWDM or colored) optical transceivers and placing an arrayed waveguide grating router (AWGR) in the center. Depending on the data center configuration and equipment prices, our colored FBFly (C-FBFly) proposal yields lower capital expenditure than the original FBFly. The key advantage of our structural modification of FBFly, however, is that in large FBFly networks (e.g., > 50K nodes) it reduces the number of inter-rack cables by a factor as large as 48

The number of centered lozenge tilings of a symmetric hexagon

Abstract. Propp conjectured [15] that the number of lozenge tilings of a semiregular hexagon of sides 2n − 1, 2n − 1 and 2n which contain the central unit rhombus is precisely one third of the total number of lozenge tilings. Motivated by this, we consider the more general situation of a semiregular hexagon of sides a, a and b. We prove explicit formulas for the number of lozenge tilings of these hexagons containing the central unit rhombus, and obtain Propp’s conjecture as a corollary of our results