Exact results for the universal area distribution of clusters in percolation, Ising and Potts models

At the critical point in two dimensions, the number of percolation clusters of enclosed area greater than A is proportional to 1/A, with a proportionality constant C that is universal. We show theoretically (based upon Coulomb gas methods), and verify numerically to high precision, that C = 1/(8 sqrt(3) pi) = 0.022972037.... We also derive, and verify to varying precision, the corresponding constant for Ising spin clusters, and for Fortuin-Kasteleyn clusters of the Q=2, 3 and 4-state Potts models.Comment: 42 pages, 9 figures, accepted for publication, J. Statis. Phy

PERFORMANCE MEASURES: BANDWIDTH VERSUS FIDELITY IN PERFORMANCE MANAGEMENT

Performance is of focal and critical interest in organizations. Despite its criticality, when it comes to human performance there are many questions as to how to best measure and manage performance. One such issue is the breadth of the performance that should be considered. In this paper, we examine the issue of the breadth of performance in terms of measuring and managing performance. Overall, a contingency approach is taken in which the expected benefits and preference for broad or narrow performance measures depend on the type of job (fixed or changeable).bandwidth, fidelity in performance management, performance measures

Purely Transmitting Defect Field Theories

We define an infinite class of integrable theories with a defect which are formulated as chiral defect perturbations of a conformal field theory. Such theories can be interacting in the bulk, and are purely transmitting through the defect. The examples of the sine-Gordon theory and Ising model are worked out in some detail.Comment: 24 pages of Tex with harvmac macros; three references have been adde

Entanglement renormalization, scale invariance, and quantum criticality

The use of entanglement renormalization in the presence of scale invariance is investigated. We explain how to compute an accurate approximation of the critical ground state of a lattice model, and how to evaluate local observables, correlators and critical exponents. Our results unveil a precise connection between the multi-scale entanglement renormalization ansatz (MERA) and conformal field theory (CFT). Given a critical Hamiltonian on the lattice, this connection can be exploited to extract most of the conformal data of the CFT that describes the model in the continuum limit.Comment: 4 pages, 3 figures, RevTeX 4. Revised for greater clarit

The Entanglement Entropy of Solvable Lattice Models

We consider the spin k/2 analogue of the XXZ quantum spin chain. We compute the entanglement entropy S associated with splitting the infinite chain into two semi-infinite pieces. In the scaling limit, we find S ~ c_k/6 (ln(xi))+ln(g)+... . Here xi is the correlation length and c_k=3k/(k+2) is the central charge associated with the sl_2 WZW model at level k. ln(g) is the boundary entropy of the WZW model. Our result extends previous observations and suggests that this is a simple and perhaps rather general way both of extracting the central charge of the ultraviolet CFT associated with the scaling limit of a solvable lattice model, and of matching lattice and CFT boundary conditions.Comment: 6 pages; connection with boundary entropy of Affleck and Ludwig added in revised version and notation slightly change

Percolation Crossing Formulas and Conformal Field Theory

Using conformal field theory, we derive several new crossing formulas at the two-dimensional percolation point. High-precision simulation confirms these results. Integrating them gives a unified derivation of Cardy's formula for the horizontal crossing probability $\Pi_h(r)$, Watts' formula for the horizontal-vertical crossing probability $\Pi_{hv}(r)$, and Cardy's formula for the expected number of clusters crossing horizontally $\mathcal{N}_h(r)$. The main step in our approach implies the identification of the derivative of one primary operator with another. We present operator identities that support this idea and suggest the presence of additional symmetry in $c=0$ conformal field theories.Comment: 12 pages, 5 figures. Numerics improved; minor correction

Shape-dependent universality in percolation

The shape-dependent universality of the excess percolation cluster number and cross-configuration probability on a torus is discussed. Besides the aspect ratio of the torus, the universality class depends upon the twist in the periodic boundary conditions, which for example are generally introduced when triangular lattices are used in simulations.Comment: 11 pages, 3 figures, to be published in Physica

Intermittency Studies in Directed Bond Percolation

The self-similar cluster fluctuations of directed bond percolation at the percolation threshold are studied using techniques borrowed from inter\-mit\-ten\-cy-related analysis in multi-particle production. Numerical simulations based on the factorial moments for large $1+1$-dimensional lattices allow to handle statistical and boundary effects and show the existence of weak but definite intermittency patterns. The extracted fractal dimensions are in agreement with scaling arguments leading to a new relation linking the intermittency indices to the critical exponents and the fractal dimension of directed percolation clusters.Comment: 16pp. Geneva preprint UGVA-DPT 1992/06/769, Saclay preprint SPhT/92-069 (to appear in Nucl. Phys. B[FS]

Difference Equations in Spin Chains with a Boundary

Correlation functions and form factors in vertex models or spin chains are known to satisfy certain difference equations called the quantum Knizhnik-Zamolodchikov equations. We find similar difference equations for the case of semi-infinite spin chain systems with integrable boundary conditions. We derive these equations using the properties of the vertex operators and the boundary vacuum state, or alternatively through corner transfer matrix arguments for the 8-vertex model with a boundary. The spontaneous boundary magnetization is found by solving such difference equations. The boundary $S$-matrix is also proposed and compared, in the sine-Gordon limit, with Ghoshal--Zamolodchikov's result. The axioms satisfied by the form factors in the boundary theory are formulated.Comment: 28 pages, LaTeX with amssymbols.sty, 7 uuencoded postscript figure

Excess number of percolation clusters on the surface of a sphere

Monte Carlo simulations were performed in order to determine the excess number of clusters b and the average density of clusters n_c for the two-dimensional "Swiss cheese" continuum percolation model on a planar L x L system and on the surface of a sphere. The excess number of clusters for the L x L system was confirmed to be a universal quantity with a value b = 0.8841 as previously predicted and verified only for lattice percolation. The excess number of clusters on the surface of a sphere was found to have the value b = 1.215(1) for discs with the same coverage as the flat critical system. Finally, the average critical density of clusters was calculated for continuum systems n_c = 0.0408(1).Comment: 13 pages, 2 figure