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Alan D. Sokal

Publications and source records attributed to Alan D. Sokal.

At least 91 records · Page 5Linked to original sources
arXiv2002

Homogeneous multivariate polynomials with the half-plane property

A polynomial P in n complex variables is said to have the "half-plane property" (or Hurwitz property) if it is nonvanishing whenever all the variables lie in the open right half-plane. Such polynomials arise in combinatorics, reliability theory, electrical circuit theory and statistical mechanics. A particularly important case is when the polynomial is homogeneous and multiaffine: then it is the (weighted) generating polynomial of an r-uniform set system. We prove that the support (set of nonzero coefficients) of a homogeneous multiaffine polynomial with the half-plane property is necessarily the set of bases of a matroid. Conversely, we ask: For which matroids M does the basis generating polynomial P_{B(M)} have the half-plane property? Not all matroids have the half-plane property, but we find large classes that do: all sixth-root-of-unity matroids, and a subclass of transversal (or cotransversal) matroids that we call "nice". Furthermore, the class of matroids with the half-plane property is closed under minors, duality, direct sums, 2-sums, series and parallel connection, full-rank matroid union, and some special cases of principal truncation, principal extension, principal cotruncation and principal coextension. Our positive results depend on two distinct (and apparently unrelated) methods for constructing polynomials with the half-plane property: a determinant construction (exploiting "energy" arguments), and a permanent construction (exploiting the Heilmann-Lieb theorem on matching polynomials). We conclude with a list of open questions.

math.CO↗
arXiv2002

Numerical Computation of \prod_{n=1}^\infty (1 - tx^n)

I present and analyze a quadratically convergent algorithm for computing the infinite product \prod_{n=1}^\infty (1 - tx^n) for arbitrary complex t and x satisfying |x| < 1, based on the identity \prod_{n=1}^\infty (1 - tx^n) = \sum_{m=0}^\infty {(-t)^m x^{m(m+1)/2} \over (1-x)(1-x^2) ... (1-x^m)} due to Euler. The efficiency of the algorithm deteriorates as |x| \uparrow 1, but much more slowly than in previous algorithms. The key lemma is a two-sided bound on the Dedekind eta function at pure imaginary argument, η(iy), that is sharp at the two endpoints y=0,\infty and is accurate to within 9.1% over the entire interval 0 < y < \infty.

math.NA↗
arXiv2001

Unusual corrections to scaling in the 3-state Potts antiferromagnet on a square lattice

At zero temperature, the 3-state antiferromagnetic Potts model on a square lattice maps exactly onto a point of the 6-vertex model whose long-distance behavior is equivalent to that of a free scalar boson. We point out that at nonzero temperature there are two distinct types of excitation: vortices, which are relevant with renormalization-group eigenvalue 1/2; and non-vortex unsatisfied bonds, which are strictly marginal and serve only to renormalize the stiffness coefficient of the underlying free boson. Together these excitations lead to an unusual form for the corrections to scaling: for example, the correlation length diverges as β\equiv J/kT \to \infty according to ξ\sim A e^{2β} (1 + bβe^{-β} + ...), where b is a nonuniversal constant that may nevertheless be determined independently. A similar result holds for the staggered susceptibility. These results are shown to be consistent with the anomalous behavior found in the Monte Carlo simulations of Ferreira and Sokal.

cond-mat.stat-mech↗
arXiv2001

Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models I. General Theory and Square-Lattice Chromatic Polynomial

We study the chromatic polynomials (= zero-temperature antiferromagnetic Potts-model partition functions) P_G(q) for m \times n rectangular subsets of the square lattice, with m \le 8 (free or periodic transverse boundary conditions) and n arbitrary (free longitudinal boundary conditions), using a transfer matrix in the Fortuin-Kasteleyn representation. In particular, we extract the limiting curves of partition-function zeros when n \to\infty, which arise from the crossing in modulus of dominant eigenvalues (Beraha-Kahane-Weiss theorem). We also provide evidence that the Beraha numbers B_2,B_3,B_4,B_5 are limiting points of partition-function zeros as n \to\infty whenever the strip width m is \ge 7 (periodic transverse b.c.) or \ge 8 (free transverse b.c.). Along the way, we prove that a noninteger Beraha number (except perhaps B_{10}) cannot be a chromatic root of any graph.

cond-mat.stat-mech↗
arXiv2000

Bounds on the Complex Zeros of (Di)Chromatic Polynomials and Potts-Model Partition Functions

I show that there exist universal constants $C(r) < \infty$ such that, for all loopless graphs $G$ of maximum degree $\le r$, the zeros (real or complex) of the chromatic polynomial $P_G(q)$ lie in the disc $|q| < C(r)$. Furthermore, $C(r) \le 7.963906... r$. This result is a corollary of a more general result on the zeros of the Potts-model partition function $Z_G(q, {v_e})$ in the complex antiferromagnetic regime $|1 + v_e| \le 1$. The proof is based on a transformation of the Whitney-Tutte-Fortuin-Kasteleyn representation of $Z_G(q, {v_e})$ to a polymer gas, followed by verification of the Dobrushin-Kotecký-Preiss condition for nonvanishing of a polymer-model partition function. I also show that, for all loopless graphs $G$ of second-largest degree $\le r$, the zeros of $P_G(q)$ lie in the disc $|q| < C(r) + 1$. Along the way, I give a simple proof of a generalized (multivariate) Brown-Colbourn conjecture on the zeros of the reliability polynomial for the special case of series-parallel graphs.

cond-mat.stat-mech↗
arXiv2000

On the chromatic roots of generalized theta graphs

The generalized theta graph Θ_{s_1,...,s_k} consists of a pair of endvertices joined by k internally disjoint paths of lengths s_1,...,s_k \ge 1. We prove that the roots of the chromatic polynomial $pi(Θ_{s_1,...,s_k},z) of a k-ary generalized theta graph all lie in the disc |z-1| \le [1 + o(1)] k/\log k, uniformly in the path lengths s_i. Moreover, we prove that Θ_{2,...,2} \simeq K_{2,k} indeed has a chromatic root of modulus [1 + o(1)] k/\log k. Finally, for k \le 8 we prove that the generalized theta graph with a chromatic root that maximizes |z-1| is the one with all path lengths equal to 2; we conjecture that this holds for all k.

math.CO↗
arXiv2000

Pathologies of the large-N limit for RP^{N-1}, CP^{N-1}, QP^{N-1} and mixed isovector/isotensor sigma-models

We compute the phase diagram in the N\to\infty limit for lattice RP^{N-1}, CP^{N-1} and QP^{N-1} sigma-models with the quartic action, and more generally for mixed isovector/isotensor models. We show that the N=\infty limit exhibits phase transitions that are forbidden for any finite N. We clarify the origin of these pathologies by examining the exact solution of the one-dimensional model: we find that there are complex zeros of the partition function that tend to the real axis as N\to\infty. We conjecture the correct phase diagram for finite N as a function of the spatial dimension d. Along the way, we prove some new correlation inequalities for a class of N-component sigma-models, and we obtain some new results concerning the complex zeros of confluent hypergeometric functions.

hep-lat↗
arXiv2000

A Personal List of Unsolved Problems Concerning Lattice Gases and Antiferromagnetic Potts Models

I review recent results and unsolved problems concerning the hard-core lattice gas and the q-coloring model (antiferromagnetic Potts model at zero temperature). For each model, I consider its equilibrium properties (uniqueness/nonuniqueness of the infinite-volume Gibbs measure, complex zeros of the partition function) and the dynamics of local and nonlocal Monte Carlo algorithms (ergodicity, rapid mixing, mixing at complex fugacity). These problems touch on mathematical physics, probability, combinatorics and theoretical computer science.

cond-mat.stat-mech↗
arXiv1999

Chromatic Polynomials, Potts Models and All That

The q-state Potts model can be defined on an arbitrary finite graph, and its partition function encodes much important information about that graph, including its chromatic polynomial, flow polynomial and reliability polynomial. The complex zeros of the Potts partition function are of interest both to statistical mechanicians and to combinatorists. I give a pedagogical introduction to all these problems, and then sketch two recent results: (a) Construction of a countable family of planar graphs whose chromatic zeros are dense in the whole complex q-plane except possibly for the disc |q-1| < 1. (b) Proof of a universal upper bound on the q-plane zeros of the chromatic polynomial (or antiferromagnetic Potts-model partition function) in terms of the graph's maximum degree.

cond-mat.stat-mech↗
arXiv1999

Antiferromagnetic Potts Models on the Square Lattice: A High-Precision Monte Carlo Study

We study the antiferromagnetic q-state Potts model on the square lattice for q=3 and q=4, using the Wang-Swendsen-Kotecky (WSK) Monte Carlo algorithm and a powerful finite-size-scaling extrapolation method. For q=3 we obtain good control up to correlation length $ξ\sim 5000$; the data are consistent with $ξ(β) = A e^{2β} β^p (1 + a_1 e^{-β} + ...)$ as $β\to\infty$, with $p \approx 1$. The staggered susceptibility behaves as $χ_{stagg} \sim ξ^{5/3}$. For q=4 the model is disordered ($ξ\ltapprox 2$) even at zero temperature. In appendices we prove a correlation inequality for Potts antiferromagnets on a bipartite lattice, and we prove ergodicity of the WSK algorithm at zero temperature for Potts antiferromagnets on a bipartite lattice.

cond-mat.stat-mech↗
arXiv1999

Universal Amplitude Ratios in the Critical Two-Dimensional Ising Model on a Torus

Using results from conformal field theory, we compute several universal amplitude ratios for the two-dimensional Ising model at criticality on a symmetric torus. These include the correlation-length ratio x^\star = \lim_{L\to\infty} ξ(L)/L and the first four magnetization moment ratios V_{2n} = <{\cal M}^{2n}>/<{\cal M}^2>^n. As a corollary we get the first four renormalized 2n-point coupling constants for the massless theory on a symmetric torus, G_{2n}^*. We confirm these predictions by a high-precision Monte Carlo simulation.

cond-mat.stat-mech↗
arXiv1996

Multi-Grid Monte Carlo via $XY$ Embedding. II. Two-Dimensional $SU(3)$ Principal Chiral Model

We carry out a high-precision simulation of the two-dimensional $SU(3)$ principal chiral model at correlation lengths $ξ$ up to $\sim 4 \times 10^5$, using a multi-grid Monte Carlo (MGMC) algorithm and approximately one year of Cray C-90 CPU time. We extrapolate the finite-volume Monte Carlo data to infinite volume using finite-size-scaling theory, and 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 $\approx 12%$ at $ξ\sim 25$, decreases to $\approx 2%$ at $ξ\sim 4 \times 10^5$. We also analyze the dynamic critical behavior of the MGMC algorithm using lattices up to $256 \times 256$, finding the dynamic critical exponent $z_{int,{\cal M}^2} \approx 0.45 \pm 0.02$ (subjective 68% confidence interval). Thus, for this asymptotically free model, critical slowing-down is greatly reduced compared to local algorithms, but not completely eliminated.

hep-lat↗
arXiv1996

Logarithmic Corrections and Finite-Size Scaling in the Two-Dimensional 4-State Potts Model

We analyze the scaling and finite-size-scaling behavior of the two-dimensional 4-state Potts model. We find new multiplicative logarithmic corrections for the susceptibility, in addition to the already known ones for the specific heat. We also find additive logarithmic corrections to scaling, some of which are universal. We have checked the theoretical predictions at criticality and off criticality by means of high-precision Monte Carlo data.

hep-lat↗
arXiv1996

Multi-Grid Monte Carlo via $XY$ Embedding I. General Theory and Two-Dimensional $O(N)$-Symmetric Nonlinear $σ$-Models

We introduce a variant of the multi-grid Monte Carlo (MGMC) method, based on the embedding of an $XY$ model into the target model, and we study its mathematical properties for a variety of nonlinear $σ$-models. We then apply the method to the two-dimensional $O(N)$-symmetric nonlinear $σ$-models (also called $N$-vector models) with $N=3,4,8$ and study its dynamic critical behavior. Using lattices up to $256 \times 256$, we find dynamic critical exponents $z_{int,{\cal M}^2} \approx 0.70 \pm 0.08$, $0.60 \pm 0.07$, $0.52 \pm 0.10$ for $N=3,4,8$, respectively (subjective 68\% confidence intervals). Thus, for these asymptotically free models, critical slowing-down is greatly reduced compared to local algorithms, but not completely eliminated; and the dynamic critical exponent does apparently vary with $N$. We also analyze the static data for $N=8$ using a finite-size-scaling extrapolation method. The correlation length $ξ$ agrees with the four-loop asymptotic-freedom prediction to within $\approx 1\%$ over the interval $12 < ξ< 650$.

hep-lat↗
arXiv1996

Absence of Phase Transition for Antiferromagnetic Potts Models via the Dobrushin Uniqueness Theorem

We prove that the $q$-state Potts antiferromagnet on a lattice of maximum coordination number $r$ exhibits exponential decay of correlations uniformly at all temperatures (including zero temperature) whenever $q > 2r$. We also prove slightly better bounds for several two-dimensional lattices: square lattice (exponential decay for $q \ge 7$), triangular lattice ($q \ge 11$), hexagonal lattice ($q \ge 4$), and Kagomé lattice ($q \ge 6$). The proofs are based on the Dobrushin uniqueness theorem.

cond-mat↗
arXiv1996

Asymptotic scaling in the two-dimensional $SU(3)$ $σ$-model at correlation length $4 \times 10^5$

We carry out a high-precision simulation of the two-dimensional $SU(3)$ principal chiral model at correlation lengths $ξ$ up to $\approx\! 4 \times 10^5$, using a multi-grid Monte Carlo (MGMC) algorithm. We extrapolate the finite-volume Monte Carlo data to infinite volume using finite-size-scaling theory, and we discuss carefully the systematic and statistical errors in this extrapolation. We then compare the extrapolated data to the renormalization-group predictions. For $ξ\gtapprox 10^3$ we observe good asymptotic scaling in the bare coupling; at $ξ\approx 4 \times 10^5$ the nonperturbative constant is within 2--3\% of its predicted limiting value.

hep-lat↗