Irreducibility of Recombination Markov Chains in the Triangular Lattice

In the United States, regions are frequently divided into districts for the purpose of electing representatives. How the districts are drawn can affect who's elected, and drawing districts to give an advantage to a certain group is known as gerrymandering. It can be surprisingly difficult to detect gerrymandering, but one algorithmic method is to compare a current districting plan to a large number of randomly sampled plans to see whether it is an outlier. Recombination Markov chains are often used for this random sampling: randomly choose two districts, consider their union, and split this union in a new way. This works well in practice, but the theory behind it remains underdeveloped. For example, it's not known if recombination Markov chains are irreducible, that is, if recombination moves suffice to move from any districting plan to any other. Irreducibility of recombination Markov chains can be formulated as a graph problem: for a graph $G$, is the space of all partitions of $G$ into $k$ connected subgraphs ($k$ districts) connected by recombination moves? We consider three simply connected districts and district sizes $k_1\pm 1$ vertices, $k_2\pm 1$ vertices, and $k3\pm 1$ vertices. We prove for arbitrarily large triangular regions in the triangular lattice, recombination Markov chains are irreducible. This is the first proof of irreducibility under tight district size constraints for recombination Markov chains beyond small or trivial examples.Comment: 79 pages, 37 figures. 10-page conference version published in SIAM Conference on Applied and Computational Discrete Algorithms, 2023 (ACDA23

A Cost Analysis and Policy Review of Digestate when deemed a Waste Product and a Fertilizer.

Food waste is a multifaceted issue that has proposed solutions as complex as the problem itself. New York State recently announced a food waste ban, effective January of 2020, that will require large scale food waste producers to manage the waste through alternative methods as opposed to disposal in a landfill. One of those methods is through Anaerobic Digestion, a process in which organic matter chemically reacts with bacteria to produce a biofuel along with an associated byproduct, namely digestate. The biofuel in most cases is used to produce electricity to feed back to the grid acting as a net benefit, but the management strategies and economics of digestate are variable. Digestate is a material high in nutrients beneficial to soil health, making it a viable option to use as a fertilizer. The matter of whether usage of digestate as a fertilizer is a net benefit or cost for the process of Anaerobic Digestion is not well known at the current state of research as assumptions are often made for this value. Scenarios comparing digestate management as a fertilizer against when it is deemed as a waste product was the main premise of the model. Research in this work will determine what the net benefit or cost of digestate is in different usage scenarios. Policies affecting the processing steps of digestate in each of the use cases are also reflected on and related to the economic analysis conducted. Digestate was found to either pose as a net benefit or cost in the fertilizer scenarios and always was a net cost in the waste management scenario. The overall goal of conducting this research is to provide key information for a solution to the overarching problem of food waste

A Local Stochastic Algorithm for Separation in Heterogeneous Self-Organizing Particle Systems

We present and rigorously analyze the behavior of a distributed, stochastic algorithm for separation and integration in self-organizing particle systems, an abstraction of programmable matter. Such systems are composed of individual computational particles with limited memory, strictly local communication abilities, and modest computational power. We consider heterogeneous particle systems of two different colors and prove that these systems can collectively separate into different color classes or integrate, indifferent to color. We accomplish both behaviors with the same fully distributed, local, stochastic algorithm. Achieving separation or integration depends only on a single global parameter determining whether particles prefer to be next to other particles of the same color or not; this parameter is meant to represent external, environmental influences on the particle system. The algorithm is a generalization of a previous distributed, stochastic algorithm for compression (PODC \u2716) that can be viewed as a special case of separation where all particles have the same color. It is significantly more challenging to prove that the desired behavior is achieved in the heterogeneous setting, however, even in the bichromatic case we focus on. This requires combining several new techniques, including the cluster expansion from statistical physics, a new variant of the bridging argument of Miracle, Pascoe and Randall (RANDOM \u2711), the high-temperature expansion of the Ising model, and careful probabilistic arguments

Fast and Perfect Sampling of Subgraphs and Polymer Systems

We give an efficient perfect sampling algorithm for weighted, connected induced subgraphs (or graphlets) of rooted, bounded degree graphs. Our algorithm utilizes a vertex-percolation process with a carefully chosen rejection filter and works under a percolation subcriticality condition. We show that this condition is optimal in the sense that the task of (approximately) sampling weighted rooted graphlets becomes impossible in finite expected time for infinite graphs and intractable for finite graphs when the condition does not hold. We apply our sampling algorithm as a subroutine to give near linear-time perfect sampling algorithms for polymer models and weighted non-rooted graphlets in finite graphs, two widely studied yet very different problems. This new perfect sampling algorithm for polymer models gives improved sampling algorithms for spin systems at low temperatures on expander graphs and unbalanced bipartite graphs, among other applications

Polynomial Mixing of the Edge-Flip Markov Chain for Unbiased Dyadic Tilings

We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall, and Spencer in 2002. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2^{-s}, (a+1)2^{-s}] x [b2^{-t}, (b+1)2^{-t}] for a,b,s,t nonnegative integers. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n^{4.09}), which implies that the mixing time is at most O(n^{5.09}). We complement this by showing that the relaxation time is at least Omega(n^{1.38}), improving upon the previously best lower bound of Omega(n*log n) coming from the diameter of the chain

Long-term stability of fibre-optic transmission for multi-object spectroscopy

We present an analysis of the long-term stability of fibre-optic transmission properties for fibre optics in astronomy. Data from 6 years of operation of the AAOmega multi-object spectrograph at the Anglo-Australian Telescope are presented. We find no evidence for significant degradation in the bulk transmission properties of the 38-m optical fibre train. Significant losses (<20 per cent relative, 4 per cent absolute) are identified and associated with the end termination of the optical fibres in the focal plane. Improved monitoring and maintenance can rectify the majority of this performance degradation