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The stable continuum Mayer pressure series converges in an explicit dilute disk

Statement

Assume the Axiom of Countable Choice. All displayed integrals use Lebesgue measure. All indices $n, N$ below range over $N$ . Fix $d \in N$ with $d \geq 1$ , an even Borel pair potential $v : R^{d} \to R \cup {+ \infty}$ with no $- \infty$ values, stable with constant $B \in R$ , $B \geq 0$ , and inverse parameter $β \in R$ with $β > 0$ . Put $e^{- c (+ \infty)} = 0$ for every $c > 0$ , $f_{β} (x) = e^{- β v (x)} - 1$ , and $C_{β} = \int_{R^{d}} ∣ f_{β} (x) ∣ d x < \infty$ . For each $n \in N$ with $n \geq 2$ , use

$U_{n} (x_{1}, \dots, x_{n}) : = \sum_{G \in C_{n}} \prod_{{i, j} \in E (G)} f_{β} (x_{i} - x_{j}),$

where $C_{n}$ is the set of connected simple graphs on ${1, \dots, n}$ ; set $U_{1} = 1$ . For $L \in R$ with $L > 0$ , let $Λ_{L} = [- L, L]^{d}$ and write $∣ Λ_{L} ∣ : = λ_{d} (Λ_{L}) = (2 L)^{d}$ ; define

$Ξ_{L} (z) : = \sum_{N = 0}^{\infty} \frac{z^{N}}{N!} \int_{Λ_{L}^{N}} \exp ⁣ (- β \sum_{1 \leq i < j \leq N} v (x_{i} - x_{j})) d x_{1} \dots d x_{N},$

with the $N = 0$ term equal to $1$ . Set $b_{1} (L) = 1$ and, for $n \geq 2$ ,

$b_{n} (L) : = \frac{1}{n! ∣ Λ_{L} ∣} \int_{Λ_{L}^{n}} U_{n} (x_{1}, \dots, x_{n}) d x_{1} \dots d x_{n} .$

Also set $b_{1} = 1$ and, for $n \geq 2$ ,

$b_{n} : = \frac{1}{n!} \int_{(R^{d})^{n - 1}} U_{n} (0, x_{2}, \dots, x_{n}) d x_{2} \dots d x_{n} .$

Let $D = {z \in C : ∣ z ∣ e^{2 β B + 1} C_{β} < 1}$ when $C_{β} > 0$ , and let $D = C$ when $C_{β} = 0$ . On $D$ , the finite-volume connected pressure series $p_{L} (z) : = \sum_{n \geq 1} b_{n} (L) z^{n}$ and the limiting Mayer pressure series $p (z) : = \sum_{n \geq 1} b_{n} z^{n}$ converge absolutely and locally uniformly, are differentiable term by term, and satisfy

$Ξ_{L} (z) = \exp ⁣ (∣ Λ_{L} ∣ p_{L} (z)), \lim_{L \to \infty} p_{L} (z) = p (z) .$

The convergence and derivative limits are uniform on every compact subdisc of $D$ . The pressure limit here is along centered cubes only.

Statement references

Stability, superstability and regular tails

Stable continuum Mayer coefficients obey an integrated tree bound

Facts & Assumptions

Given: The potential, stability constant, inverse parameter, Mayer integrability, and Countable Choice stated above.

[F1]

Stability, superstability and regular tails: $v$ is even Borel with values in $R \cup {+ \infty}$ , and stability means $\sum_{i < j} v (x_{i} - x_{j}) \geq - B N$ for every finite list of $N$ positions. Borel measurability means that preimages of target Borel sets are domain Borel sets, with $B (R^{d})$ generated by Euclidean open sets and $B (\overline{R})$ the order Borel sigma-algebra (The Borel sigma-algebra of a topological space, The Borel sigma-algebra on the extended real line).

[F2]

Stable continuum Mayer coefficients obey an integrated tree bound: each $U_{n}$ is real-valued, $∣ U_{n} (x_{1}, \dots, x_{n}) ∣ \leq e^{2 β B n} \sum_{T \in T_{n}} \prod_{{i, j} \in E (T)} ∣ f_{β} (x_{i} - x_{j}) ∣$ , and the pinned integral is at most $e^{2 β B n} n^{n - 2} C_{β}^{n - 1}$ ; graph components are maximal path-connected induced subgraphs, so every finite graph decomposes into a unique partition of its vertices into connected components; the tree count $n^{n - 2}$ is proved by Prüfer words.

[F3]

The Axiom of Countable Choice ( ${A C}_{ω}$ ) and Lebesgue measure is sigma-finite, and every metrically bounded subset of $R^{n}$ has finite outer measure: under Countable Choice, Lebesgue measure is sigma-finite and finite on bounded sets.

[F4]

Dominated convergence for the measure integral: a.e. convergence under an integrable majorant permits passage of the limit through the integral.

[F5]

Fubini's theorem for L^1 functions on a sigma-finite product: absolutely integrable functions on sigma-finite product spaces may be integrated in either order.

[F6]

Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation: Lebesgue measure and measurability are invariant under translations.

[F7]

The real exponential function and the number $e$ by a power series and The exponential addition formula $\exp (x + y) = \exp (x) \exp (y)$ : $e^{x} = \sum_{k \geq 0} x^{k} / k!$ and $e^{x + y} = e^{x} e^{y}$ on $R$ .

[F8]

The complex numbers as $R [x] / (x^{2} + 1)$ , with the real embedding and imaginary unit $i$ , The complex exponential by its power series, and $\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential: the complex exponential is defined by its everywhere convergent power series and satisfies $\exp (u + v) = \exp (u) \exp (v)$ .

[F9]

Inside its disc of convergence a complex power series is holomorphic and may be differentiated term by term: a complex power series differentiates term by term strictly inside its radius of convergence.

[F10]

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product: nonnegative product-measurable functions on sigma-finite product spaces may be integrated in either order.

[F11]

Composition with a Borel measurable outer map preserves measurability and Arithmetic and lattice operations preserve measurability whenever they are defined: Borel composition and finite arithmetic preserve measurability wherever the operations are defined.

[F12]

The exponential function is strictly increasing: the real exponential is continuous and increasing.

[F13]

Continuous functions on Euclidean spaces are Borel measurable: under Countable Choice, continuous maps between Euclidean spaces are Borel measurable.

[F14]

Extended-real-valued measurable functions and The Borel sigma-algebra on the extended real line: the extended real line carries its order Borel sigma-algebra, and ${+ \infty}$ is Borel.

[F15]

The Borel product of R^m and R^n is the Borel sigma-algebra of R^{m+n}: for two Euclidean factors, the product Borel sigma-algebra equals the Borel sigma-algebra of the product Euclidean space.

[F16]

Assuming countable choice, every Borel subset of $R^{n}$ is Lebesgue measurable: under Countable Choice, every Borel subset of $R^{d}$ is Lebesgue measurable.

[F17]

A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included: every box in $R^{d}$ has its product-of-side-lengths measure, including $∣ [- L, L]^{d} ∣ = (2 L)^{d}$ .

[F18]

The modulus of an integral is bounded by the integral of the modulus: for an integrable function, the modulus of its integral is at most the integral of its modulus.

[F19]

Monotonicity and nonnegative homogeneity of the nonnegative integral: the nonnegative integral is monotone and homogeneous under nonnegative scalar multiplication.

[F20]

Additivity of the nonnegative Lebesgue integral: the integral of a sum of nonnegative measurable functions is the sum of their integrals.

[F21]

Real and imaginary parts, complex conjugation, and modulus and Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive: $∣ x + i y ∣ = \sqrt{x^{2} + y^{2}}$ ; complex modulus is nonnegative and multiplicative. The canonical real-embedding identity is stated in Complex modulus restricts to real absolute value. Its current imported carrier is marked published and has provenance.proof: ai-generated, but records only verification.precheck: pass, with no verification.audited or verification.verified; the format pass does not evidence proof acceptance.

[F22]

Absolute value in an ordered field, Basic properties of the absolute value: real absolute value is defined piecewise by sign and is nonnegative and multiplicative.

[F23]

The sum $\sum_{i \in S} a_{i}$ over a finite index set, and its product form: sums over finite sets are enumeration-independent finite sums, with empty sum $0$ .

[A1]

Assume Countable Choice, as required for sigma-finite Lebesgue products, Tonelli and Fubini.

Proof

Given: [F1], [F2], [F3], [F4], [F5], [F6], [F7], [F8], [F9], [F10], [F11], [F12], [F13], [F14], [F15], [F16], [F17], [F18], [F19], [F20], [A1], and the definitions in the Statement.

1.1A1F1F3F7F11F12F13F14F15F16F17F21givenalgebra

For every $N$ -point configuration, the energy $E_{N} = \sum_{i < j} v (x_{i} - x_{j})$ is an extended-real Borel function: coordinate differences are continuous and hence Borel [F13], composition with $v$ preserves Borel measurability [F11], and finite sums preserve it wherever defined. The map $u \mapsto e^{- β u}$ on the extended real line, assigned $0$ at $+ \infty$ and $+ \infty$ at $- \infty$ , is Borel because it is continuous on $R$ [F12] and the endpoints are Borel [F14]. Thus the Boltzmann integrand is Borel on $R^{d N}$ . For $N \geq 2$ , repeated application of the two-factor product identity [F15] identifies the finite product Borel sigma-algebra; the Borel-to-Lebesgue inclusion [F16] then makes the integrand product-measurable. For $N = 0$ or $1$ the same measurability is immediate from the empty or one-factor product. The same extended exponential map and the Borel potential show that $f_{β}$ is Borel. Stability [F1] gives $E_{N} \geq - B N$ , so the Boltzmann integrand is real, nonnegative and at most $e^{β B N}$ , with a forbidden $+ \infty$ configuration contributing $0$ . Since $∣ Λ_{L} ∣ = (2 L)^{d} < \infty$ [F17], multiplicativity of complex modulus and its agreement with real absolute value [F21] bound the modulus of the $N$ th term of $Ξ_{L} (z)$ by $(∣ z ∣ e^{β B} ∣ Λ_{L} ∣)^{N} / N!$ . The sum of these bounds is finite for every $z$ by [F7], so $Ξ_{L}$ is entire.

1.2A1F1F2F5F11F12F17F23givenalgebra

Stability at $N = 2$ gives $v (x) \geq - 2 B$ for every $x \in R^{d}$ ; hence $- 1 \leq f_{β} (x) \leq e^{2 β B} - 1$ by [F12]. Thus for each fixed $N$ every graph term in the finite expansion $\prod_{i < j} (1 + f_{β} (x_{i} - x_{j}))$ is bounded and measurable on the finite-measure box $Λ_{L}^{N}$ . Expanding the product gives a sum over all simple graphs; each graph decomposes uniquely into connected components as in [F2]. If the component sizes are $m_{1}, \dots, m_{k}$ , assigning the $N$ labels to an ordered list of blocks contributes $N! / \prod_{j} m_{j}!$ , and division by $k!$ removes block order. Integrating a product over disjoint coordinate blocks factorizes by Fubini [F5], since the factors are bounded on finite-measure boxes. Therefore the coefficient of $z^{N}$ in $Ξ_{L}$ equals that in the formal exponential of $A_{L} (z) : = \sum_{n \geq 1} \frac{z^{n}}{n!} \int_{Λ_{L}^{n}} U_{n} (x_{1}, \dots, x_{n}) d x_{1} \dots d x_{n}$ .

1.3A1F2F3F6F10F18F19F20F21F22F17givenalgebra

For $n \geq 2$ , integrate the pointwise tree estimate [F2] over $Λ_{L}^{n}$ . Root each tree at vertex $1$ , then remove leaves other than the root. At each removal the leaf integral over $Λ_{L}$ is at most the whole-space integral $C_{β}$ by monotonicity [F19] and translation invariance [F6]; Tonelli [F10] applies to the nonnegative tree products. The final root integral is $∣ Λ_{L} ∣$ . The labelled-tree count in [F2] is $n^{n - 2}$ . Finite additivity [F20] sums those tree integrals, giving $\int_{Λ_{L}^{n}} ∣ U_{n} ∣ \leq e^{2 β B n} ∣ Λ_{L} ∣ n^{n - 2} C_{β}^{n - 1}$ . Thus $U_{n}$ is integrable on $Λ_{L}^{n}$ . Regard its real values as complex values by the real embedding; [F21] identifies the complex modulus with real absolute value, so the integral triangle inequality [F18] gives $∣ b_{n} (L) ∣ \leq e^{2 β B n} n^{n - 2} C_{β}^{n - 1} / n!$ . The same bound holds for $∣ b_{n} ∣$ by the pinned estimate [F2].

1.4A1F2F3F4F5F6F10F11F13F14F15F16F17given

Fix $n \geq 2$ and write $y_{1} = 0$ , $y_{i} = x_{i} - x_{1}$ for $i \geq 2$ . Translation invariance of $U_{n}$ gives $U_{n} (x_{1}, \dots, x_{n}) = U_{n} (0, y_{2}, \dots, y_{n})$ . By Tonelli [F10], translation invariance [F6], and the pinned integrability from [F2], $\int_{Λ_{L}^{n}} ∣ U_{n} (x) ∣ d x = \int_{(R^{d})^{n - 1}} ∣ U_{n} (0, y) ∣ ∣ Λ_{L} \cap (Λ_{L} - y_{2}) \cap \dots \cap (Λ_{L} - y_{n}) ∣ d y \leq ∣ Λ_{L} ∣ \int ∣ U_{n} (0, y) ∣ d y < \infty$ , using the finite cube volume [F17]. Thus the integrand is absolutely integrable and Fubini [F5] applies. Translating each remaining coordinate in its section gives $b_{n} (L) = \frac{1}{n!} \int_{(R^{d})^{n - 1}} U_{n} (0, y_{2}, \dots, y_{n}) θ_{L} (y) d y_{2} \dots d y_{n}$ , where $θ_{L} (y) = \frac{∣ Λ_{L} \cap (Λ_{L} - y_{2}) \cap \dots \cap (Λ_{L} - y_{n}) ∣}{∣ Λ_{L} ∣} = \prod_{j = 1}^{d} (1 - \frac{\max_{i} y_{i, j} - \min_{i} y_{i, j}}{2 L})_{+}$ , with $r_{+} = \max (r, 0)$ . For each fixed $y$ , $0 \leq θ_{L} (y) \leq 1$ and $θ_{L} (y) \to 1$ . The cube-intersection volume formula follows from [F17]; $U_{n}$ is Borel by [F11, F13, F14, F15, F16], hence measurable. For every sequence $L_{m} \to \infty$ , dominated convergence [F4] with majorant $∣ U_{n} (0, \cdot) ∣$ gives $b_{n} (L_{m}) \to b_{n}$ . If the limit as a real variable $L \to \infty$ failed, some $ϵ > 0$ would have $∣ b_{n} (L) - b_{n} ∣ \geq ϵ$ for arbitrarily large $L$ ; Countable Choice [A1] selects such an $L_{m} \geq m$ for each $m$ , contradicting the sequential conclusion. Hence $b_{n} (L) \to b_{n}$ .

2.1F2F7F9F21step 1.3algebra

For every integer $n \geq 2$ , the positive series for $e^{n}$ in [F7] contains the term $n^{n} / n!$ , so $n^{n - 2} / n! \leq e^{n} / n^{2}$ . If $C_{β} > 0$ , put $q (r) = r C_{β} e^{2 β B + 1}$ . For $0 < r$ with $q (r) < 1$ , step 1.3 yields $∣ b_{n} (L) ∣ r^{n} \leq r e^{2 β B + 1} q (r)^{n - 1} / n^{2}$ and $n ∣ b_{n} (L) ∣ r^{n - 1} \leq e^{2 β B + 1} q (r)^{n - 1} / n$ ; the same bounds hold for $b_{n}$ . Both majorant series converge. If $C_{β} = 0$ , step 1.3 gives $b_{n} (L) = b_{n} = 0$ for $n \geq 2$ . Each $U_{n}$ is real-valued by [F2], so each coefficient $b_{n} (L), b_{n}$ is real; [F21] identifies its complex-embedding modulus with its real absolute value and gives $∣ z^{n} ∣ = ∣ z ∣^{n}$ . Hence the coefficient and derivative majorants bound the complex power-series terms on the corresponding closed subdiscs. Thus the pressure series and their formal derivative series converge uniformly on every closed subdisc strictly inside $D$ ; [F9] identifies the derivatives with the termwise derivatives.

3.1F8F9step 1.1step 1.2step 1.4step 2.1∎

The coefficient identity in step 1.2 is the formal identity $Ξ_{L} = \exp (A_{L})$ , and $A_{L} (z) = ∣ Λ_{L} ∣ p_{L} (z)$ by the definitions of $b_{n} (L)$ . Fix a volume $L > 0$ and a radius $r$ strictly inside $D$ . Step 2.1 gives $\sum_{n} ∣ b_{n} (L) ∣ r^{n} < \infty$ , so $\sum_{n} ∣ [z^{n}] A_{L} ∣ r^{n} = ∣ Λ_{L} ∣ \sum_{n} ∣ b_{n} (L) ∣ r^{n} < \infty$ for this fixed volume. Expanding the complex exponential in [F8], the sum of absolute values of all terms is bounded by $\exp (\sum_{n} ∣ [z^{n}] A_{L} ∣ r^{n})$ ; hence its products and coefficient sums may be regrouped, so the formal identity is an equality of convergent series and $Ξ_{L} (z) = \exp (∣ Λ_{L} ∣ p_{L} (z))$ . To pass to the limit on a compact subdisc $∣ z ∣ \leq r$ inside $D$ , the pressure and derivative tails are uniformly bounded in $L$ by the convergent majorants of step 2.1. For each fixed $n$ , step 1.4 gives coefficient convergence; finite partial sums and their derivatives therefore converge uniformly, and the uniform tail bounds give $p_{L} \to p$ and $p_{L}^{'} \to p^{'}$ uniformly there.

This result gives a sufficient dilute analyticity domain. It does not assert liquid-vapor coexistence, identify a physical phase boundary, or establish pressure limits along arbitrary shapes.

Remarks