Automatic Proofs for Formulae Enumerating Proper Polycubes

This video describes a general framework for computing formulae enumerating polycubes of size n which are proper in n-k dimensions (i.e., spanning all n-k dimensions), for a fixed value of k. (Such formulae are central in the literature of statistical physics in the study of percolation processes and collapse of branched polymers.) The implemented software re-affirmed the already-proven formulae for k <= 3, and proved rigorously, for the first time, the formula enumerating polycubes of size n that are proper in n-4 dimensions

Counting Lattice Animals in High Dimensions

We present an implementation of Redelemeier's algorithm for the enumeration of lattice animals in high dimensional lattices. The implementation is lean and fast enough to allow us to extend the existing tables of animal counts, perimeter polynomials and series expansion coefficients in $d$-dimensional hypercubic lattices for $3 \leq d\leq 10$. From the data we compute formulas for perimeter polynomials for lattice animals of size $n\leq 11$ in arbitrary dimension $d$. When amended by combinatorial arguments, the new data suffices to yield explicit formulas for the number of lattice animals of size $n\leq 14$ and arbitrary $d$. We also use the enumeration data to compute numerical estimates for growth rates and exponents in high dimensions that agree very well with Monte Carlo simulations and recent predictions from field theory.Comment: 18 pages, 7 figures, 6 tables; journal versio

Logarithmic observables in critical percolation

Although it has long been known that the proper quantum field theory description of critical percolation involves a logarithmic conformal field theory (LCFT), no direct consequence of this has been observed so far. Representing critical bond percolation as the Q = 1 limit of the Q-state Potts model, and analyzing the underlying S_Q symmetry of the Potts spins, we identify a class of simple observables whose two-point functions scale logarithmically for Q = 1. The logarithm originates from the mixing of the energy operator with a logarithmic partner that we identify as the field that creates two propagating clusters. In d=2 dimensions this agrees with general LCFT results, and in particular the universal prefactor of the logarithm can be computed exactly. We confirm its numerical value by extensive Monte-Carlo simulations.Comment: 11 pages, 2 figures. V2: as publishe

DNFSB recommendation 94-1 Hanford site integrated stabilization management plan

In May 1994, the Defense Nuclear Facilities Safety Board (DNFSB) issued DNFSB Recommendation 94-1 (Conway 1994), which identified concerns related to US Department of Energy (DOE) management of legacy fissile materials remaining from past defense production activities. The DNFSB expressed concern about the existing storage conditions for these materials and the slow pace at which the conditions were being remediated. The DNFSB also expressed its belief that additional delays in stabilizing these fissile materials would be accompanied by further deterioration of safety and unnecessary increased risks to workers and the public. In February 1995, DOE issued the DNFSB Recommendation 94-1 Implementation Plan (O`Leary 1995) to address the concerns identified in DNFSB Recommendation 94-1. The Implementation Plan (IP) identifies several DOE commitments to achieve safe interim storage for the legacy fissile materials, and constitutes DOE`s baseline DNFSB Recommendation 94-1 Integrated Program Plan (IPP). The IPP describes the actions DOE plans to implement within the DOE complex to convert its excess fissile materials to forms or conditions suitable for safe interim storage. The IPP was subsequently supplemented with an Integrated Facilities Plan and a Research and Development Plan, which further develop complex-wide research and development and long-range facility requirements and plans. The additions to the baseline IPP were developed based on a systems engineering approach that integrated facilities and capabilities at the various DOE sites and focused on attaining safe interim storage with minimum safety risks and environmental impacts. Each affected DOE site has developed a Site Integrated Stabilization Management Plan (SISMP) to identify individual site plans to implement the DNFSB Recommendation 94-1 IPP. The SISMPs were developed based on the objectives, requirements, and commitments identified in the DNFSB Recommendation 94-1 IP. The SISMPs also supported formulation of the initial versions of the Integrated Facilities Plan and the Research and Development Plan. The SISMPs are periodically updated to reflect improved integration between DOE sites as identified during the IPP systems engineering evaluations. This document is the fifth update of the Hanford SISMP

The puzzle of bulk conformal field theories at central charge c=0

Non-trivial critical models in 2D with central charge c=0 are described by Logarithmic Conformal Field Theories (LCFTs), and exhibit in particular mixing of the stress-energy tensor with a "logarithmic" partner under a conformal transformation. This mixing is quantified by a parameter (usually denoted b), introduced in [V. Gurarie, Nucl. Phys. B 546, 765 (1999)], and which was first thought to play the role of an "effective" central charge. The value of b has been determined over the last few years for the boundary versions of these models: $b_{\rm perco}=-5/8$ for percolation and $b_{\rm poly} = 5/6$ for dilute polymers. Meanwhile, the existence and value of $b$ for the bulk theory has remained an open problem. Using lattice regularization techniques we provide here an "experimental study" of this question. We show that, while the chiral stress tensor has indeed a single logarithmic partner in the chiral sector of the theory, the value of b is not the expected one: instead, b=-5 for both theories. We suggest a theoretical explanation of this result using operator product expansions and Coulomb gas arguments, and discuss the physical consequences on correlation functions. Our results imply that the relation between bulk LCFTs of physical interest and their boundary counterparts is considerably more involved than in the non-logarithmic case.Comment: 5 pages, published versio

Detection of an anomalous cluster in a network

We consider the problem of detecting whether or not, in a given sensor network, there is a cluster of sensors which exhibit an "unusual behavior." Formally, suppose we are given a set of nodes and attach a random variable to each node. We observe a realization of this process and want to decide between the following two hypotheses: under the null, the variables are i.i.d. standard normal; under the alternative, there is a cluster of variables that are i.i.d. normal with positive mean and unit variance, while the rest are i.i.d. standard normal. We also address surveillance settings where each sensor in the network collects information over time. The resulting model is similar, now with a time series attached to each node. We again observe the process over time and want to decide between the null, where all the variables are i.i.d. standard normal, and the alternative, where there is an emerging cluster of i.i.d. normal variables with positive mean and unit variance. The growth models used to represent the emerging cluster are quite general and, in particular, include cellular automata used in modeling epidemics. In both settings, we consider classes of clusters that are quite general, for which we obtain a lower bound on their respective minimax detection rate and show that some form of scan statistic, by far the most popular method in practice, achieves that same rate to within a logarithmic factor. Our results are not limited to the normal location model, but generalize to any one-parameter exponential family when the anomalous clusters are large enough.Comment: Published in at http://dx.doi.org/10.1214/10-AOS839 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org