Charles M. Newman

Tree graph inequalities and critical behavior in percolation models

Various inequalities are derived and used for the study of the critical behavior in independent percolation models. In particular, we consider the critical exponent γ associated with the expected cluster sizex and the structure of then-site connection probabilities τ=τn(x1,..., xn). It is shown that quite generally γ⩾ 1. The upper critical dimension, above which γ attains the Bethe lattice value 1, is characterized both in terms of the geometry of incipient clusters and a diagramatic convergence condition. For homogeneousd-dimensional lattices with τ(x, y)=O(¦x -y¦−(d−2+η), atp=pc, our criterion shows that γ=1 if η> (6-d)/3. The connectivity functions τn are generally bounded by tree diagrams which involve the two-point function. We conjecture that above the critical dimension the asymptotic behavior of τn, in the critical regime, is actually given by such tree diagrams modified by a nonsingular vertex factor. Other results deal with the exponential decay of the cluster-size distribution and the function τ2(x, y).

Limit Theorems for Sums of Dependent Random Variables Occurring in Statistical Mechanics

SummaryWe study the asymptotic behavior of partial sums Snfor certain triangular arrays of dependent, identically distributed random variables which arise naturally in statistical mechanics. A typical result is that under appropriate assumptions there exist a real number m, a positive real number λ, and a positive integer k so that (Sn−nm)/n1−1/2k converges weakly to a random variable with density proportional to exp(−λ¦s¦2k/(2k)!). We explain the relation of these results to topics in Gaussian quadrature, to the theory of moment spaces, and to critical phenomena in physical systems.

The statistics of Curie-Weiss models

LetSn denote the random total magnetization of ann-site Curie-Weiss model, a collection ofn (spin) random variables with an equal interaction of strength 1/n between each pair of spins. The asymptotic behavior for largen of the probability distribution ofSn is analyzed and related to the well-known (mean-field) thermodynamic properties of these models. One particular result is that at a type-k critical point (Sn-nm)/n1−1/2k has a limiting distribution with density proportional to exp[-λs2k/(2k)!], wherem is the mean magnetization per site and A is a positive critical parameter with a universal upper bound. Another result describes the asymptotic behavior relevant to metastability.

Inequalities for Ising models and field theories which obey the Lee-Yang Theorem

A series of inequalities for partition, correlation, and Ursell functions are derived as consequences of the Lee-Yang Theorem. In particular, then-point Schwinger functions ofeven φ4 models are bounded in terms of the 2-point function as strongly as is the case for Gaussian fields; this strengthens recent results of Glimm and Jaffe and shows that renormalizability of the 2-point function by fourth degree counter-terms implies existence of a φ4 field theory with a moment generating function which is entire of exponential order at most two. It is also noted that ifany (even) truncated Schwinger function vanishes identically, the resulting field theory is a generalized free field.

The GHS and Other Correlation Inequalities for a Class of Even Ferromagnets

We prove the GHS inequality for families of random variables which arise in certain ferromagnetic models of statistical mechanics and quantum field theory. These include spin −1/2 Ising models, ϕ4 field theories, and other continuous spin models. The proofs are based on the properties of a classG− of probability measures which contains all measures of the form const exp(−V(x))dx, whereV is even and continuously differentiable anddV/dx is convex on [0, ∞). A new proof of the GKS inequalities using similar ideas is also given.

Nearest neighbors and Voronoi regions in certain point processes

We investigate, for several models of point processes, the (random) number N of points which have a given point as their nearest neighbor. The largedimensional limit of Poisson processes is treated by considering N = Nd for n points independently and uniformly distributed in a d-dimensional cube of volume n and showing that lim,_, lim ,Na d Poisson (A = 1). An asymptotic Poisson (A = 1) distribution also holds for many of the other models. On the other hand, we find that limn--lim ,,, N D 0. Related results concern the (random) volume, Vol%, of a Voronoi polytope (or Dirichlet cell) in the cube model; we find that limd.oolimn-.Vol D 1 while lim_,, lim__, Vold 9 0. POISSON PROCESSES; DIRICHLET CELLS

Nearest neighbor analysis of point processes: Applications to multidimensional scaling

Abstract A new approach for evaluating spatial statistical models based on the (random) number 0 ≤ N ( i , n ) ≤ n of points whose nearest neighbor is i in an ensemble of n + 1 points is discussed. The second moment of N ( i , n ) offers a measure of the centrality of the ensemble. The asymptotic distribution of N ( i , n ) and the expected degree of centrality for several spatial and nonspatial point processes is described. The use of centrality as a diagnostic statistic for multidimensional scaling is explored.

Metastability and the analytic continuation of eigenvalues

A metastable analytic continuation of the Ising model free energy is conjectured to follow from certain analyticity properties of the eigenvalues of the transfer matrix. The resulting description of metastability is applicable to any system whose phase transition is associated with eigenvalue degeneracy. Motivation for the conjectures concerning the Ising model is provided by the study of eigenvalue continuation in several simpler systems.

Critical point inequalities and scaling limits

A refined and extended version of the Buckingham-Gunton inequality relating various pairs of critical exponents is shown to be valid for a large class of statistical mechanical models. If this inequality is an equality (in the refined sense) and one of the critical exponents has a non-Gaussian value, then any scaling limit must be non-Gaussian. This result clarifies the relationship between the nontriviality or triviality of the scaling limit for ordinary critical points in four dimensions (or tricritical points in three dimensions) and the existence of logarithmic factors in the asymptotics which define the two critical exponents.

Moment inequalities for ferromagnetic Gibbs distributions

Moment inequalities analogous to Khintchine’s inequality (for sums of independent Bernoulli random variables) are obtained for a certain class of random variables which naturally arises in the context of ferromagnetic Ising models and φ4 Euclidean (quantum) field models in a positive external field. These results extend ones obtained previously which applied only to the mean zero (vanishing external field) case.