Trees, forests and jungles: a botanical garden for cluster expansions

Combinatoric formulas for cluster expansions have been improved many times over the years. Here we develop some new combinatoric proofs and extensions of the tree formulas of Brydges and Kennedy, and test them on a series of pedagogical examples.Comment: 37 pages, Ecole Polytechnique A-325.099

A simple method for finite range decomposition of quadratic forms and Gaussian fields

We present a simple method to decompose the Green forms corresponding to a large class of interesting symmetric Dirichlet forms into integrals over symmetric positive semi-definite and finite range (properly supported) forms that are smoother than the original Green form. This result gives rise to multiscale decompositions of the associated Gaussian free fields into sums of independent smoother Gaussian fields with spatially localized correlations. Our method makes use of the finite propagation speed of the wave equation and Chebyshev polynomials. It improves several existing results and also gives simpler proofs.Comment: minor correction for t<

The Global Renormalization Group Trajectory in a Critical Supersymmetric Field Theory on the Lattice Z^3

We consider an Euclidean supersymmetric field theory in $Z^3$ given by a supersymmetric $\Phi^4$ perturbation of an underlying massless Gaussian measure on scalar bosonic and Grassmann fields with covariance the Green's function of a (stable) L\'evy random walk in $Z^3$. The Green's function depends on the L\'evy-Khintchine parameter $\alpha={3+\epsilon\over 2}$ with $0<\alpha<2$. For $\alpha ={3\over 2}$ the $\Phi^{4}$ interaction is marginal. We prove for $\alpha-{3\over 2}={\epsilon\over 2}>0$ sufficiently small and initial parameters held in an appropriate domain the existence of a global renormalization group trajectory uniformly bounded on all renormalization group scales and therefore on lattices which become arbitrarily fine. At the same time we establish the existence of the critical (stable) manifold. The interactions are uniformly bounded away from zero on all scales and therefore we are constructing a non-Gaussian supersymmetric field theory on all scales. The interest of this theory comes from the easily established fact that the Green's function of a (weakly) self-avoiding L\'evy walk in $Z^3$ is a second moment (two point correlation function) of the supersymmetric measure governing this model. The control of the renormalization group trajectory is a preparation for the study of the asymptotics of this Green's function. The rigorous control of the critical renormalization group trajectory is a preparation for the study of the critical exponents of the (weakly) self-avoiding L\'evy walk in $Z^3$.Comment: 82 pages, Tex with macros supplied. Revision includes 1. redefinition of norms involving fermions to ensure uniqueness. 2. change in the definition of lattice blocks and lattice polymer activities. 3. Some proofs have been reworked. 4. New lemmas 5.4A, 5.14A, and new Theorem 6.6. 5.Typos corrected.This is the version to appear in Journal of Statistical Physic

Fermion Determinants

The current status of bounds on and limits of fermion determinants in two, three and four dimensions in QED and QCD is reviewed. A new lower bound on the two-dimensional QED determinant is derived. An outline of the demonstration of the continuity of this determinant at zero mass when the background magnetic field flux is zero is also given.Comment: 10 page

The analyticity region of the hard sphere gas. Improved bounds

We find an improved estimate of the radius of analyticity of the pressure of the hard-sphere gas in $d$ dimensions. The estimates are determined by the volume of multidimensional regions that can be numerically computed. For $d=2$, for instance, our estimate is about 40% larger than the classical one.Comment: 4 pages, to appear in Journal of Statistical Physic

Abstract polymer models with general pair interactions

A convergence criterion of cluster expansion is presented in the case of an abstract polymer system with general pair interactions (i.e. not necessarily hard core or repulsive). As a concrete example, the low temperature disordered phase of the BEG model with infinite range interactions, decaying polynomially as $1/r^{d+\lambda}$ with $\lambda>0$, is studied.Comment: 19 pages. Corrected statement for the stability condition (2.3) and modified section 3.1 of the proof of theorem 1 consistently with (2.3). Added a reference and modified a sentence at the end of sec. 2.

Abstract cluster expansion with applications to statistical mechanical systems

We formulate a general setting for the cluster expansion method and we discuss sufficient criteria for its convergence. We apply the results to systems of classical and quantum particles with stable interactions

Rooted Spiral Trees on Hyper-cubical lattices

We study rooted spiral trees in 2,3 and 4 dimensions on a hyper cubical lattice using exact enumeration and Monte-Carlo techniques. On the square lattice, we also obtain exact lower bound of 1.93565 on the growth constant $\lambda$. Series expansions give $\theta=-1.3667\pm 0.001$ and $\nu = 1.3148\pm0.001$ on a square lattice. With Monte-Carlo simulations we get the estimates as $\theta=-1.364\pm0.01$, and $\nu = 1.312\pm0.01$. These results are numerical evidence against earlier proposed dimensional reduction by four in this problem. In dimensions higher than two, the spiral constraint can be implemented in two ways. In either case, our series expansion results do not support the proposed dimensional reduction.Comment: replaced with published versio

Hard squares with negative activity

We show that the hard-square lattice gas with activity z= -1 has a number of remarkable properties. We conjecture that all the eigenvalues of the transfer matrix are roots of unity. They fall into groups (``strings'') evenly spaced around the unit circle, which have interesting number-theoretic properties. For example, the partition function on an M by N lattice with periodic boundary condition is identically 1 when M and N are coprime. We provide evidence for these conjectures from analytical and numerical arguments.Comment: 8 page

On the convergence of cluster expansions for polymer gases

We compare the different convergence criteria available for cluster expansions of polymer gases subjected to hard-core exclusions, with emphasis on polymers defined as finite subsets of a countable set (e.g. contour expansions and more generally high- and low-temperature expansions). In order of increasing strength, these criteria are: (i) Dobrushin criterion, obtained by a simple inductive argument; (ii) Gruber-Kunz criterion obtained through the use of Kirkwood-Salzburg equations, and (iii) a criterion obtained by two of us via a direct combinatorial handling of the terms of the expansion. We show that for subset polymers our sharper criterion can be proven both by a suitable adaptation of Dobrushin inductive argument and by an alternative --in fact, more elementary-- handling of the Kirkwood-Salzburg equations. In addition we show that for general abstract polymers this alternative treatment leads to the same convergence region as the inductive Dobrushin argument and, furthermore, to a systematic way to improve bounds on correlations