Alan D. Sokal

An improvement of Watson’s theorem on Borel summability

Watson’s theorem, which gives sufficient conditions for Borel summability, is not optimal. Watson assumes analyticity and uniform asymptotic expansion in a sector ‖argz‖<π/2+e, ‖z‖<R, with e≳0; in fact, only the circular region Re(1/z) ≳1/R is required. In particular, one can take e=0. This improved theorem gives a necessary and sufficient characterization of a large class of Borel‐summable functions. I apply it to the perturbation expansion in the φ24 quantum field theory.

New Monte Carlo method for the self-avoiding walk

We introduce a new Monte Carlo algorithm for the self-avoiding walk (SAW), and show that it is particularly efficient in the critical region (long chains). We also introduce new and more efficient statistical techniques. We employ these methods to extract numerical estimates for the critical parameters of the SAW on the square lattice. We findμ=2.63820 ± 0.00004 ± 0.00030γ=1.352 ± 0.006 ± 0.025νv=0.7590 ± 0.0062 ± 0.0042 where the first error bar represents systematic error due to corrections to scaling (subjective 95% confidence limits) and the second bar represents statistical error (classical 95% confidence limits). These results are based on SAWs of average length ≈ 166, using 340 hours CPU time on a CDC Cyber 170–730. We compare our results to previous work and indicate some directions for future research.

A general Lee-Yang theorem for one-component and multicomponent ferromagnets

We show that any measure on ℝn possessing the Lee-Yang property retains that property when multiplied by a ferromagnetic pair interaction. Newmans Lee-Yang theorem for one-component ferromagnets with general single-spin measure is an immediate consequence. We also prove an analogous result for two-component ferromagnets. ForN-component ferromagnets (N ≧ 3), we prove a Lee-Yang theorem when the interaction is sufficiently anisotropic.

A new proof of the existence and nontriviality of the continuum ϕ 2 4 and ϕ 3 4 quantum field theories

We use Schwinger-Dyson equations combined with rigorous “perturbation-theoretic” correlation inequalities to give a new and extremely simple proof of the existence and nontriviality of the weakly-coupled continuum ϕ24 and ϕ34 quantum field theories, constructed as subsequence limits of lattice theories. We prove an asymptotic expansion to order λ or λ2 for the correlation functions and for the mass gap. All Osterwalder-Schrader axioms are satisfied except perhaps Euclidean (rotation) invariance.

MORE INEQUALITIES FOR CRITICAL EXPONENTS

A variety of rigorous inequalities for critical exponents is proved. Most notable is the low-temperature Josephson inequalitydv′ ⩾γ′+2β ⩾ 2−α′. Others are 1 ⩽γ′ ⩽ 1 +v′φ, 1 ⩽ζ ⩽ 1δμφ,δ ⩾ 1,dμφ ⩾ 1 + 1/δ (for φ ⩾d),dv′φ, ⩾ Δ′3 + α (for φ ⩾d), Δ4 ⩾γ, and Δ2m ⩽ Δ2m+2 (form ⩾ 2). The hypotheses vary; all inequalities are true for the spin-1/2 Ising model with nearest-neighbor ferromagnetic pair interactions.

A rigorous inequality for the specific heat of an Ising or φ4 ferromagnet

Abstract For a class of models including the Ising ferromagnet and the φ 4 lattice field theory, I prove a rigorous upper bound for the specific heat in terms of the susceptibility and the magnetization. This bound implies the critical-exponent inequalities α ⩽ max (0,( 2 − d 2 )γ), α′ ⩽ max (0,( 2 − d 2 )γ′, γ′ − 2β), and α c ⩽ max (0, ( 2 − d 2 ) (δ − 1), δ − 3) for lattice dimension d ⩾ 2.

Existence of compatible families of proper regular conditional probabilities

SummaryLet (Ω, ℱ, μ) be a perfect probability space with ℱ countably generated, and let IB be a family of sub-σ-fields of ℱ. Under a countability condition on the family IB, I show that there exists a family {π∇}∇∈IB of regular conditional probabilities which are everywhere compatible. Under a more stringent condition on IB, I show that the π∇ can furthermore be chosen to be everywhere proper. It follows that in the Dobrushin-Lanford-Ruelle formulation of the statistical mechanics of classical lattice systems, every (perfect) probability measure is a Gibbs measure for some specification.

Mean-field bounds and correlation inequalities

I prove a new correlation inequality for a class ofN-component classical ferromagnets (1⩽N⩽4). This inequality implies that the correlation functions decay exponentially and the spontaneous magnetization is zero, above the mean-field critical temperature.

More surprises in the general theory of lattice systems

I use Israels methods to prove new theorems of “ubiquitous pathology” for classical and quantum lattice systems. The main result is the following: Let Φ be any interaction and ϱ be any translation-invariant equilibrium state for Φ (extremal or not). Then there exists a sequence {Φk} of interactions converging to Φ, having extremal (or even unique) translation-invariant equilibrium states ϱk, such that {ϱk} converges to ϱ. In certain situations the perturbations Φk−Φ can be chosen to lie in a cone of “antiferromagnetic pair interactions.” I discuss the connection with results of Daniëls and van Enter, and point out an application to the one-dimensional ferromagnetic Ising model with 1/r2 interaction (Thouless effect).

Scaling in multichain polymer systems in two and three dimensions

The mean dimensions of multichain polymer systems are predicted to follow a scaling relation with scaling variable X=ldν−1 ρ, where l is the number of statistical segments on the chain, ρ is the segment density, d is the dimension, and ν is the critical exponent for the mean dimensions of an isolated polymer chain. The scaling laws are 〈R2〉≊A(X) l2ν for l→ ∞ with X bounded, and 〈R2〉≊B(ρ) l for l→ ∞ with X → ∞. Moreover, the critical amplitudes behave as A(X)∼X−(2ν−1)/(dν−1) as X → ∞ and B(ρ)∼ρ−(2ν−1)/(dν−1) as ρ → 0. Simulations of both continuum and lattice systems are reanalyzed and found to be consistent with these scaling relations. Previous naive use of short‐chain data has led to misleading results.