A Multi-Body Dobrushin-Sokal Criterion -- Part II

Technical Summary of "A Multi-Body Dobrushin-Sokal Criterion – Part II"

Problem Statement
This paper addresses the convergence of Mayer cluster expansions and polymer expansions for classical lattice gases featuring complex-valued, multi-body interactions. While extensive literature exists for pairwise interactions (e.g., Kotecký–Preiss, Dobrushin–Sokal, Fernández–Procacci conditions), extending these criteria to higher-order interactions presents significant combinatorial challenges. Specifically, the Kirkwood–Salsburg hierarchy, a systematic tool for deriving convergence criteria, becomes considerably more complex when applied to multi-body potentials due to non-trivial diagrammatic interpretations. Previous approaches often relied on restrictive assumptions, such as finite-range interactions or stable pairwise potentials augmented by higher-order terms, or yielded estimates that were "somewhat crude" compared to the pairwise case. The paper aims to derive a sufficient condition for the absolute convergence of these expansions without assuming finite-range interactions, applicable to lattice gases with site-wise hard-core self-repulsion and general complex multi-body interactions.

Methodology
The author employs a systematic approach combining the Kirkwood–Salsburg hierarchy with a novel combinatorial tool. The core methodology involves:

1. Kirkwood–Salsburg Hierarchy: The paper utilizes the hierarchy to relate partition functions and correlation functions. Unlike previous works that treated the kernel combinatorics as trivial (in the pairwise case) or used crude estimates, this work tackles the diagrammatic interpretation of the kernel directly.
2. Partition Scheme for Spanning Hypergraphs: A central technical innovation is Lemma 4.1, which introduces a partition scheme for spanning hypergraphs. This lemma allows for the decomposition of the set of hypergraphs with a specific span into disjoint subsets defined by a mapping $m$ and a maximal extension $M$. This scheme replaces intricate isolation techniques used in recent works on hypergraph independence polynomials (e.g., Galvin et al., Bencs and Buys) with a more direct insertion into the Kirkwood–Salsburg hierarchy.
3. Ruelle's Algebraic Method: The framework relies on "Ruelle's algebraic method" to manipulate power series, utilizing Poisson convolution, $\ast$-exponentials, and $\ast$-logarithms. This allows for the rigorous handling of formal power series for partition functions ($Z$) and Mayer series ($M$).
4. Inductive Bounds: The proofs proceed by establishing inductive bounds on effective activities ($\hat{b}_{\Lambda}$) and correlation functions. The author defines auxiliary functions $\alpha$ and $r$ (where $r = \alpha/(1+\alpha)$) to bound the absolute values of these quantities, ensuring they remain within convergence radii.

Key Contributions and Results

• Theorem 2.1 (Main Result): The paper establishes a sufficient condition for the absolute convergence of the Mayer cluster expansion for lattice gases with multi-body interactions. The condition requires that for every site $x$, a specific inequality involving the activity $z(x)$, the interaction potential $V$, and an auxiliary function $\alpha$ holds:
  $$|z(x)| \prod_{X \Subset X: x \in X} \max\left\{ |e^{-V(X)}|, 1 + |f(X)|\alpha_{S} \mid \emptyset \neq S \subset X \setminus \{x\} \right\} \leq r(x)$$
  where $f = e^{-V} - 1$ is the Mayer function.

  • Implications: Under this condition, the effective activity is bounded by $r(x) < 1$, and the correlation functions are bounded by $\alpha(x) < \infty$. This guarantees the absolute convergence of the Mayer series and the non-vanishing of the partition function for $r$-finite volumes.
  • Generality: This result subsumes classical pairwise results (Dobrushin–Sokal, Kotecký–Preiss) and extends them to multi-body cases, including higher-order hard-core repulsions.

• Theorem 2.4 (Polymer Expansion): The paper extends the analysis to the polymer expansion, where the lattice gas is reformulated in terms of non-overlapping "polymers" (finite subsets of sites). The author derives a convergence condition (Theorem 2.4) that utilizes a transformation of the activity $z \to z/(1+z)$.

  • Improvement: This approach yields convergence conditions that extend to parameter ranges beyond those covered by Theorem 2.1, particularly for "physical" activities. It provides notable improvements over conditions found in Procacci and Scoppola [22] and Gallavotti, Miracle-Solè, and Robinson [9].

• Corollary 2.5: By specializing the potential to scale with an inverse temperature $\beta$, the paper derives explicit high-temperature criteria. This corollary improves upon existing results in [22] and [25] by providing tighter bounds on the activity region where phase transitions are absent.

• Comparison with Nguyen and Fernández (2024): The author compares the derived criteria with recent results by Nguyen and Fernández [6]. While Nguyen and Fernández's method applies to broader lattice systems, the author notes that their method covers stronger interactions, including hard-core ones, by dropping the requirement that the auxiliary function $A \geq 1$.

Significance and Claims
The paper claims to provide a "multi-body Dobrushin-Sokal criterion" that is:

1. Systematic: It leverages the Kirkwood–Salsburg hierarchy to provide a unified approach to convergence criteria, avoiding overly technical a priori assumptions.
2. Optimal in Specific Regimes: By bridging the gap between the results of Galvin et al. and Bencs and Buys (who worked on hypergraph independence polynomials) and classical lattice gas theory, the method achieves optimal uniform radii in terms of global degree bounds for hard-core repulsions.
3. Versatile: The framework handles complex-valued interactions and higher-order hard-core repulsions, which are difficult to treat with previous pairwise-centric methods.
4. Complementary to Part I: While Part I focused on the non-vanishing of partition functions, this Part II explicitly states and proves the convergence of the cluster expansions, completing the theoretical framework.

The author modestly notes that while the method is well-suited for stronger multi-body interactions, it does not necessarily apply to the most general continuum systems or all possible lattice spin systems without reformulation, acknowledging that other recent preprints (e.g., Helmuth, Pappik, and Perkins) address different classes of systems. The primary contribution is the derivation of a robust, single-site inequality condition that ensures absolute convergence for a wide class of multi-body lattice gases.