Abstract Cluster Expansion with Application to Statistical Mechanical Systems

CLUSTER EXPANSION WITH APPLICATIONS 3 Assumption 2. There exists a nonnegative function a on X such that for almost all x ∈ X, ∫ d|μ|(y) |ζ(x, y)| e 6 a(x). In order to guess the correct form of a, one should consider the left side of the equation above with a(y) ≡ 0. The integral may depend on x; a typical situation is that x is characterized by a length l(x), which is a positive number, so that the left side is roughly proportional to l(x). This suggests to try a(x) = cl(x), and one can then optimize on the value of c. We also consider an alternate criterion that involves u rather than ζ. We use it in Section 5.2 when studying a system of quantum particles that interact via an integrable stable potential. Assumption 2’. There exists a nonnegative function a on X such that for almost all x ∈ X, ∫ d|μ|(y) |u(x, y)| e 6 a(x). For positive u we can take b(x) ≡ 0; and since 1 − e 6 u, Assumption 2 is always better than Assumption 2’. But Assumption 2’ has its advantages too; notice also that it involves b(y) and not 2b(y). We actually conjecture that, together with Assumption 1, a sufficient condition is ∫ d|μ|(y) min ( |ζ(x, y)|, |u(x, y)| ) e 6 a(x). (2.5) That is, it should be possible to combine the best of both assumptions. But this remains to be proved. We denote by Gn the set of all graphs with n vertices (unoriented, no loops) and Cn ⊂ Gn the set of connected graphs with n vertices. We introduce the following combinatorial function on finite sequences (x1, . . . , xn) of elements of X: φ(x1, . . . , xn) = { 1 if n = 1, 1 n! ∑ G∈Cn ∏ {i,j}∈G ζ(xi, xj) if n > 2. (2.6) The product is over edges of G. Theorem 2.1 (Cluster expansions). Suppose that Assumptions 1 and 2, or 1 and 2’, hold true. We also suppose that ∫ d|μ|(y)| e < ∞. Then we have Z = exp { ∑