Correction-to-scaling exponents for two-dimensional self-avoiding walks

Abstract

We study the correction-to-scaling exponents for the two-dimensional self-avoiding walk, using a combination of series-extrapolation and Monte Carlo methods. We enumerate all self-avoiding walks up to 59 steps on the square lattice, and up to 40 steps on the triangular lattice, measuring the mean-square end-to-end distance, the mean-square radius of gyration and the mean-square distance of a monomer from the endpoints. The complete endpoint distribution is also calculated for self-avoiding walks up to 32 steps (square) and up to 22 steps (triangular). We also generate self-avoiding walks on the square lattice by Monte Carlo, using the pivot algorithm, obtaining the mean-square radii to ~0.01% accuracy up to N = 4000. We give compelling evidence that the first non-analytic correction term for two-dimensional self-avoiding walks is Delta_1 = 3/2. We compute several moments of the endpoint distribution function, finding good agreement with the field-theoretic predictions. Finally, we study a particular invariant ratio that can be shown, by conformal-field-theory arguments, to vanish asymptotically, and we find the cancellation of the leading analytic correction.

arXiv commentLaTeX 2.09, 56 pages. Version 2 adds a renormalization-group discussion near the end of Section 2.2, and makes many small improvements in the exposition. To be published in the Journal of Statistical Physics

References in corpus (21)

• Critical Phenomena and Renormalization-Group Theory
• Critical Exponents, Hyperscaling and Universal Amplitude Ratios for Two- and Three-Dimensional Self-Avoiding Walks
• 25th-order high-temperature expansion results for three-dimensional Ising-like systems on the simple cubic lattice
• Statistics of lattice animals (polyominoes) and polygons
• Enumeration of self-avoiding walks on the square lattice
• Self-avoiding polygons on the square lattice
• Universal Properties of Self-Avoiding Walks from Two-Dimensional Field Theory
• A parallel algorithm for the enumeration of self-avoiding polygons on the square lattice
• The susceptibility of the square lattice Ising model: New developments
• Algebraic Techniques for Enumerating Self-Avoiding Walks on the Square Lattice
• The two-point correlation function of three-dimensional O(N) models: critical limit and anisotropy
• Universal Amplitude Combinations for Self-Avoiding Walks, Polygons and Trails
• Irrelevant operators in the two-dimensional Ising model
• A strong-coupling analysis of two-dimensional O(N) sigma models with N<=2 on square, triangular and honeycomb lattices
• Self-avoiding walks and polygons on the triangular lattice
• Critical amplitudes and mass spectrum of the 2D Ising model in a magnetic field
• Determination of the exponent gamma for SAWs on the two-dimensional Manhattan lattice
• End-to-end distribution function for dilute polymers
• Enumeration of self avoiding trails on a square lattice using a transfer matrix technique
• Corrections to scaling in 2--dimensional polymer statistics
• Osculating and neighbour-avoiding polygons on the square lattice

Cited by in corpus (27)

• Minimally subtracted six loop renormalization of $O(n)$-symmetric ${\varphi }^4$ theory and critical exponents
• Efficient implementation of the pivot algorithm for self-avoiding walks
• Calculation of the connective constant for self-avoiding walks via the pivot algorithm
• Shape anisotropy of polymers in disordered environment
• Two-dimensional self-avoiding walks and polymer adsorption: Critical fugacity estimates
• Series extension: Predicting approximate series coefficients from a finite number of exact coefficients
• Corrections to Scaling for Watersheds, Optimal Path Cracks, and Bridge Lines
• Honeycomb lattice polygons and walks as a test of series analysis techniques
• Polymer adsorption on a fractal substrate: numerical study
• Critical Loop Gases and the Worm Algorithm
• Solution of a model of SAW's with multiple monomers per site on the Husimi lattice
• Scaling Analysis of Random Walks with Persistence Lengths: Application to Self-Avoiding Walks
• Exact sampling of self-avoiding paths via discrete Schramm-Loewner evolution
• Endless self-avoiding walks
• Theta-point polymers in the plane and Schramm-Loewner evolution
• Pressure exerted by a grafted polymer on the limiting line of a semi-infinite square lattice
• Square lattice self-avoiding walks and biased differential approximants
• Extrapolation from hypergeometric functions, continued functions and Borel-Leroy transformation; Resummation of perturbative renormalization functions from field theories
• A numerical adaptation of SAW identities from the honeycomb to other 2D lattices
• Two-dimensional interacting self-avoiding walks: new estimates for critical temperatures and exponents
• Structural characterization of ice polymorphs from self-avoiding walks
• Continued functions and critical exponents: Tools for analytical continuation of divergent expressions in phase transition studies
• Numerical estimates of square lattice star vertex exponents
• Critical behaviour of the extended-ballistic transition for pulled self-avoiding walks
• Critical Exponents of the O(N)-symmetric ${\varphi }^4$ Model from the \boldmath{\large{ ${\epsilon }^7$}} Hypergeometric-Meijer Resummation
• On the shape of Gaussian scale-free polymer networks
• Entropic exponents of grafted lattices stars