Large N and the renormalization group

In the large N limit, we show that the Local Potential Approximation to the flow equation for the Legendre effective action, is in effect no longer an approximation, but exact - in a sense, and under conditions, that we determine precisely. We explain why the same is not true for the Polchinski or Wilson flow equations and, by deriving an exact relation between the Polchinski and Legendre effective potentials (that holds for all N), we find the correct large N limit of these flow equations. We also show that all forms (and all parts) of the renormalization group are exactly soluble in the large N limit, choosing as an example, D dimensional O(N) invariant N-component scalar field theory.Comment: 13 pages, uses harvmac; Added: one page with further clarification of the main results, discussion of earlier work, and new references. To be published in Phys. Lett.

Ribosomal footprints on a transcriptome landscape

Next-generation sequencing technology can be used to assess rates and regulation of translation across the entire yeast transcriptome

Derivative expansion of the renormalization group in O(N) scalar field theory

We apply a derivative expansion to the Legendre effective action flow equations of O(N) symmetric scalar field theory, making no other approximation. We calculate the critical exponents eta, nu, and omega at the both the leading and second order of the expansion, associated to the three dimensional Wilson-Fisher fixed points, at various values of N. In addition, we show how the derivative expansion reproduces exactly known results, at special values N=infinity,-2,-4, ... .Comment: 29 pages including 4 eps figures, uses LaTeX, epsfig, and latexsy

The Boundary Cosmological Constant in Stable 2D Quantum Gravity

We study further the r\^ole of the boundary operator \O_B for macroscopic loop length in the stable definition of 2D quantum gravity provided by the $[{\tilde P},Q]=Q$ formulation. The KdV flows are supplemented by an additional flow with respect to the boundary cosmological constant $\sigma$. We numerically study these flows for the $m=1$, $2$ and $3$ models, solving for the string susceptibility in the presence of \O_B for arbitrary coupling $\sigma$. The spectrum of the Hamiltonian of the loop quantum mechanics is continuous and bounded from below by $\sigma$. For large positive $\sigma$, the theory is dominated by the `universal' $m=0$ topological phase present only in the $[{\tilde P},Q]=Q$ formulation. For large negative $\sigma$, the non--perturbative physics approaches that of the $[P,Q]=1$ definition, although there is no path to the unstable solutions of the $[P,Q]=1$ $m$-even models.Comment: (Plain Tex, 11pp, 4 figures available on request) SHEP 91/92-2

A statnet Tutorial

The statnet suite of R packages contains a wide range of functionality for the statistical analysis of social networks, including the implementation of exponential-family random graph (ERG) models. In this paper we illustrate some of the functionality of statnet through a tutorial analysis of a friendship network of 1,461 adolescents.

ergm: A Package to Fit, Simulate and Diagnose Exponential-Family Models for Networks

We describe some of the capabilities of the ergm package and the statistical theory underlying it. This package contains tools for accomplishing three important, and inter-related, tasks involving exponential-family random graph models (ERGMs): estimation, simulation, and goodness of fit. More precisely, ergm has the capability of approximating a maximum likelihood estimator for an ERGM given a network data set; simulating new network data sets from a fitted ERGM using Markov chain Monte Carlo; and assessing how well a fitted ERGM does at capturing characteristics of a particular network data set.

statnet: Software Tools for the Representation, Visualization, Analysis and Simulation of Network Data

statnet is a suite of software packages for statistical network analysis. The packages implement recent advances in network modeling based on exponential-family random graph models (ERGM). The components of the package provide a comprehensive framework for ERGM-based network modeling, including tools for model estimation, model evaluation, model-based network simulation, and network visualization. This broad functionality is powered by a central Markov chain Monte Carlo (MCMC) algorithm. The coding is optimized for speed and robustness.

Specification of Exponential-Family Random Graph Models: Terms and Computational Aspects

Exponential-family random graph models (ERGMs) represent the processes that govern the formation of links in networks through the terms selected by the user. The terms specify network statistics that are sufficient to represent the probability distribution over the space of networks of that size. Many classes of statistics can be used. In this article we describe the classes of statistics that are currently available in the ergm package. We also describe means for controlling the Markov chain Monte Carlo (MCMC) algorithm that the package uses for estimation. These controls affect either the proposal distribution on the sample space used by the underlying Metropolis-Hastings algorithm or the constraints on the sample space itself. Finally, we describe various other arguments to core functions of the ergm package

Specification of Exponential-Family Random Graph Models: Terms and Computational Aspects

Exponential-family random graph models (ERGMs) represent the processes that govern the formation of links in networks through the terms selected by the user. The terms specify network statistics that are sufficient to represent the probability distribution over the space of networks of that size. Many classes of statistics can be used. In this article we describe the classes of statistics that are currently available in the ergm package. We also describe means for controlling the Markov chain Monte Carlo (MCMC) algorithm that the package uses for estimation. These controls affect either the proposal distribution on the sample space used by the underlying Metropolis-Hastings algorithm or the constraints on the sample space itself. Finally, we describe various other arguments to core functions of the ergm package