The Ne Plus Ultra of Tree Graph Inequalities*

With nary mention of a tree graph, we obtain a cluster expansion bound that includes and vastly generalizes bounds as obtained by extant tree graph inequalities. This includes applications to both two-body and many-body potential situations of the recently obtained 'new improved tree graph inequalities" that have led to the 'extra I/N! factors'. We work in a formalism coupling a discrete set of boson variables, such as occurs in a lattice system in classical statistical mechanics, or in Euclidean quantum field theory. The estimates of this Letter apply to numerical factors as arising in cluster expansions, due to essentially arbitrary sequences of the basic operations: interpolation of the covariance, interpolation of the interaction, and integration by parts. This includes complicated evolutions, such as the repeated use of interpolation to decouple the same variables several times, to ensure higher connectivity for renormalization purposes, in quantum field theory. AMS subject classifications (1980). 82A67, 81El0. Tree graph inequalities are intimately involved in cluster expansion estimation. The earliest tree graph inequalities were developed by Gl imm and Jaffe in applications to quan tum field theory. But, in fact, they have impor tant applications in simpler situations, such as in the study of the Mayer expansion in classical statistical mechanics. Perhaps the simplest cluster expansion (in the mos t leisurely presentation), using tree graph estimation, is given in [3], developing the Mayer expansion in a classical gas with two-body interactions. (See also [5], page 40.) In [1], an improved method o f tree graph estimation was developed. ( In this paper [ 1], the case o f two-body interactions is treated in Section 8, and the case o f many-body interactions in Appendix B. But it should be noted that in cluster expansions involving many-body interactions, one can often organize the combinatoric estimates, by grouping variables together, so that one only needs the same bounds as apply in the two-body case. This was true for the problem studied in [1].) With this new tree graph estimation procedure, 'Extra l /N! factors ' are obtainable in bounds; these have been impor tan t in m a n y applications. This estimation procedure has been further exposed in [2] and [4]. The moral o f this development was that to bound contr ibut ions as arising in a cluster expansion one should g roup terms with the same ' interaction structure ' together. Using a c o m m o n estimate for the interaction structure, the sum of the integrals over interpolat ion parameters adds to a simple numerical weighting. ( In fact equalling one!) One does not * This work was supported in part by the National Science Foundation under grant no. PHY-87-01329. 338 PAUL FEDERBUSH distinguish different histories of development of these terms, different orders of differentiation. Estimation thus pursued yielded the 'extra 1/N! factors', and was easy to accommodate to the 'sums to sups' procedure standard in cluster expansion study. In the cluster expansion applications discussed in the last paragraph, each term in the cluster expansion is associated to a tree graph, with each vertex associated to a subset of variables, and with the vertices ordered. Each vertex arises from an expansion step in the cluster expansion, the ordering as given by the sequence of steps, the variables as introduced at the given step. The new tree graph estimation procedure of [1, 2, 4] consists primarily of disregarding the ordering, grouping terms together associated to tree graphs with the same structure, the same variables at each vertex, but possibly different orders of vertices such terms are said to have the same 'interaction structure'. In the present Letter, we will deal with cluster expansion terms not necessarily associated to tree graph structures, but to diagrams with higher connectivity structure. In this Letter, we treat an expectation as given in (1), below, and consider it expanded as a sum of terms, in (8), below. The expansion is developed employing the unit operations of interpolation of the interaction, interpolation of the covariance, and integration by parts (the first two operations occur combined). We assume familiarity with these general operations, and explicit steps in minimal detail basically to establish notation. In developing a connected kernel in a cluster expansion, or a polymer, these operations are inductively performed according to carefully prescribed rules. The usual tree graph estimation procedure is intimately entangled with these expansion rules. Our estimate, Equation (16) below, holds for any evolution of the unit operations, essentially arbitrary interpolations, and is not naturally associated to any tree graph estimation. Such more complicated evolutions are important in the phase cell expansions of quantum field theories [1]. One may, for example, use interpolation repeatedly to decouple the same variables several times, to ensure higher connectivity for renormalization purposes. We foresee many other applications. Although the present Letter is written in a boson field formalism, estimates for a potential theory situation with two(or many-) body interactions, for example, are obtainable by considering terms developed herein by interpolation of the interaction alone (this is a trivial observation). The bounds we derive are proved by identifying the coefficients in a Taylor series, the method of proof given in [1]. This technique, not intimately tied to tree graphs, we view as more general than the tree graph oriented development in [4]. We consider the integral E E = f ~ ( d ) e -~' e Vp, (1)