arXiv subjects
Publications and source records attributed to Alan D. Sokal.
This article is a pedagogical review of Monte Carlo methods for the self-avoiding walk, with emphasis on the extraordinarily efficient algorithms developed over the past decade. Many more details can be found in hep-lat/9405016.
In this talk we present the exact solution of the most general one-dimensional $O(N)$-invariant spin model taking values in the sphere $S^{N-1}$, with nearest-neighbour interactions, and we discuss the possible continuum limits. All these results are obtained using a high-temperature expansion in terms of hyperspherical harmonics. Applications in higher dimensions of the same technique are then discussed.
We have computed the four-loop contribution to the beta-function and to the anomalous dimension of the field for the two-dimensional lattice $N$-vector model. This allows the determination of the second perturbative correction to various long-distance quantities like the correlation lengths and the susceptibilities. We compare these predictions with new Monte Carlo data for $N = 3,4,8$. From these data we also extract the values of various universal nonperturbative constants, which we compare with the predictions of the $1/N$ expansion.
We study the dynamic critical behavior of a Swendsen--Wang--type algorithm for the Ashkin--Teller model. We find that the Li--Sokal bound on the autocorrelation time ($τ_{{\rm int},{\cal E}} \ge {\rm const} \times C_H$) holds along the self-dual curve of the symmetric Ashkin--Teller model, but this bound is apparently not sharp. The ratio $τ_{{\rm int},{\cal E}}/C_H$ appears to tend to infinity either as a logarithm or as a small power ($0.05 \ltapprox p \ltapprox 0.12$).
We introduce a new and very convenient approach to multi-grid Monte Carlo (MGMC) algorithms for general nonlinear $σ$-models: it is based on embedding an $XY$ model into the given $σ$-model, and then updating the induced $XY$ model using a standard $XY$-model MGMC code. We study the dynamic critical behavior of this algorithm for the two-dimensional $O(N)$ $σ$-models with $N = 3,4,8$ and for the $SU(3)$ principal chiral model. We find that the dynamic critical exponent $z$ varies systematically between these different asymptotically free models: it is approximately 0.70 for $O(3)$, 0.60 for $O(4)$, 0.50 for $O(8)$, and 0.45 for $SU(3)$. It goes without saying that we have no theoretical explanation of this behavior.
We solve exactly the general one-dimensional $O(N)$-invariant spin model taking values in the sphere $S^{N-1}$, with nearest-neighbor interactions, in finite volume with periodic boundary conditions, by an expansion in hyperspherical harmonics. The possible continuum limits are discussed for a general one-parameter family of interactions, and an infinite number of universality classes is found. For these classes we compute the finite-size-scaling functions and the leading corrections to finite-size scaling. A special two-parameter family of interactions (which includes the mixed isovector/isotensor model) is also treated, and no additional universality classes appear. In the appendices we give new formulae for the Clebsch-Gordan coefficients and 6--$j$ symbols of the $O(N)$ group, and some new generalizations of the Poisson summation formula; these may be of independent interest.
We study the dynamic critical behavior of the multi-grid Monte Carlo (MGMC) algorithm with piecewise-constant interpolation and a W-cycle, applied to the one-dimensional $O(4)$-symmetric nonlinear $σ$-model [= $SU(2)$ principal chiral model], on lattices from $L=128$ to $L=16384$. Our data for the integrated autocorrelation time $τ_{int,{\cal M}^2}$ are well fit by a logarithmic growth. We have no idea why the critical slowing-down is not completely eliminated.
We present a simple and powerful method for extrapolating finite-volume Monte Carlo data to infinite volume, based on finite-size-scaling theory. We discuss carefully its systematic and statistical errors, and we illustrate it using three examples: the two-dimensional three-state Potts antiferromagnet on the square lattice, and the two-dimensional $O(3)$ and $O(\infty)$ $σ$-models. In favorable cases it is possible to obtain reliable extrapolations (errors of a few percent) even when the correlation length is 1000 times larger than the lattice.
We carry out a high-precision Monte Carlo simulation of the two-dimensional $O(3)$-invariant $σ$-model at correlation lengths $ξ$ up to $\sim 10^5$. Our work employs a new and powerful method for extrapolating finite-volume Monte Carlo data to infinite volume, based on finite-size-scaling theory. We compare the extrapolated data to the renormalization-group predictions. The deviation from asymptotic scaling, which is $\approx 25\%$ at $ξ\sim 10^2$, decreases to $\approx 4\%$ at $ξ\sim 10^5$.
We carry out a high-precision Monte Carlo simulation of the two-dimensional $O(3)$-invariant $σ$-model at correlation lengths $ξ$ up to $\sim 10^5$. Our work employs a new and powerful method for extrapolating finite-volume Monte Carlo data to infinite volume, based on finite-size-scaling theory. We discuss carefully the systematic and statistical errors in this extrapolation. We then compare the extrapolated data to the renormalization-group predictions. The deviation from asymptotic scaling, which is $\approx 25\%$ at $ξ\sim 10^2$, decreases to $\approx 4\%$ at $ξ\sim 10^5$.
We make a high-precision Monte Carlo study of two- and three-dimensional self-avoiding walks (SAWs) of length up to 80000 steps, using the pivot algorithm and the Karp-Luby algorithm. We study the critical exponents $ν$ and $2Δ_4 -γ$ as well as several universal amplitude ratios; in particular, we make an extremely sensitive test of the hyperscaling relation $dν= 2Δ_4 -γ$. In two dimensions, we confirm the predicted exponent $ν= 3/4$ and the hyperscaling relation; we estimate the universal ratios $\ / \ = 0.14026 \pm 0.00007$, $\ / \ = 0.43961 \pm 0.00034$ and $Ψ^* = 0.66296 \pm 0.00043$ (68\% confidence limits). In three dimensions, we estimate $ν= 0.5877 \pm 0.0006$ with a correction-to-scaling exponent $Δ_1 = 0.56 \pm 0.03$ (subjective 68\% confidence limits). This value for $ν$ agrees excellently with the field-theoretic renormalization-group prediction, but there is some discrepancy for $Δ_1$. Earlier Monte Carlo estimates of $ν$, which were $\approx\! 0.592$, are now seen to be biased by corrections to scaling. We estimate the universal ratios $\ / \ = 0.1599 \pm 0.0002$ and $Ψ^* = 0.2471 \pm 0.0003$; since $Ψ^* > 0$, hyperscaling holds. The approach to $Ψ^*$ is from above, contrary to the prediction of the two-parameter renormalization-group theory. We critically reexamine this theory, and explain where the error lies.
We present a simple and powerful method for extrapolating finite-volume Monte Carlo data to infinite volume, based on finite-size-scaling theory. We discuss carefully its systematic and statistical errors, and we illustrate it using three examples: the two-dimensional three-state Potts antiferromagnet on the square lattice, and the two-dimensional $O(3)$ and $O(\infty)$ $σ$-models. In favorable cases it is possible to obtain reliable extrapolations (errors of a few percent) even when the correlation length is 1000 times larger than the lattice.
This article is a pedagogical review of Monte Carlo methods for the self-avoiding walk, with emphasis on the extraordinarily efficient algorithms developed over the past decade.
We study the antiferromagnetic $q$-state Potts model on the square lattice for $q=3$ and $q=4$, using the Wang-Swendsen-Kotecký Monte Carlo algorithm and a new finite-size-scaling extrapolation method. For $q=3$ we obtain good control up to correlation length $ξ\sim 5000$; the data are consistent with $ξ(β) = A e^{2β} β(1 + a_1 e^{-β} + \ldots)$ as $β\to\infty$. For $q=4$ the model is disordered even at zero temperature.
We argue that the two-dimensional $O(N)$-invariant lattice $σ$-model with mixed isovector/isotensor action has a one-parameter family of nontrivial continuum limits, only one of which is the continuum $σ$-model constructed by conventional perturbation theory. We test the proposed scenario with a high-precision Monte Carlo simulation for $N=3,4$ on lattices up to $512 \times 512$, using a Wolff-type embedding algorithm. [CPU time $\approx$ 7 years IBM RS-6000/320H] The finite-size-scaling data confirm the existence of the predicted new family of continuum limits. In particular, the $RP^{N-1}$ and $N$-vector models do not lie in the same universality class.
I make three conceptual points regarding multigrid methods for computing propagators in lattice gauge theory: 1) The class of operators handled by the algorithm must be stable under coarsening. 2) Problems related by symmetry should have solution methods related by symmetry. 3) It is crucial to distinguish the vector space $V$ from its dual space $V^*$. All the existing algorithms violate one or more of these principles.
We prove that for any Monte Carlo algorithm of Metropolis type, the autocorrelation time of a suitable ``energy''-like observable is bounded below by a multiple of the corresponding ``specific heat''. This bound does not depend on whether the proposed moves are local or non-local; it depends only on the distance between the desired probability distribution $π$ and the probability distribution $π^{(0)}$ for which the proposal matrix satisfies detailed balance. We show, with several examples, that this result is particularly powerful when applied to non-local algorithms.
Previous theories of dilute polymer solutions have failed to distinguish clearly between two very different ways of taking the long-chain limit: (I) $N \to\infty$ at fixed temperature $T$, and (II) $N \to\infty$, $T \to T_θ$ with $x \equiv N^ϕ(T-T_θ)$ fixed. I argue that the modern two-parameter theory (continuum Edwards model) applies to case II --- not case I --- and in fact gives exactly the crossover scaling functions for $x \ge 0$ modulo two nonuniversal scale factors. A Wilson-type renormalization group clarifies the connection between crossover scaling functions and continuum field theories. [Also contains a general discussion of the connection between the Wilson and field-theoretic renormalization groups. Comments solicited.]