A component proof of Ursell-function monotonicity

The current revision of our Lee–Yang paper, with Jui-Hui Chung and Jacob Shapiro, contains a self-contained proof of Ursell-function monotonicity. This supplies the missing input for the shorter correlation-based proof of our gap comparison.

What is monotone?

Ursell functions are joint cumulants: they isolate the connected part of a many-spin correlation. At zero field, their even orders have alternating signs. On any finite ferromagnetic Ising graph, write

$$U_{2k}(x_1,\ldots,x_{2k})=(-1)^{k-1}u_{2k}(x_1,\ldots,x_{2k}).$$

The statement is that $U_{2k}\geq0$ and

$$\frac{\partial U_{2k}}{\partial J_e}\geq0$$

for every nonnegative edge coupling $J_e$, including when marked sites repeat. Thus strengthening a ferromagnetic interaction increases the signed connected correlation.

Camia, Jiang and Newman stated the monotonicity theorem in 2023. Our revision gives a different proof. It also records a counterexample to an auxiliary restricted-sign claim used in their argument—not to the monotonicity theorem itself.

Remove a component, then induct

The proof begins with two independent copies of the Ising model. Their half-sum and half-difference separate the sites where the spins agree from those where they disagree. Expanding the disagreement interaction as a random current, we group terms by the connected component containing a marked site.

Removing that component leaves a strictly smaller graph. The coupling derivative of a magnetization becomes a sum of magnetizations on these smaller graphs, multiplied by nonnegative current weights and ratios of partition functions. This component identity is the key reduction.

Further field derivatives produce two kinds of factors: signed Ursell functions on the smaller graph, and differences between signed Ursell functions on the original and smaller graphs. A double induction closes the argument: the first factors are controlled by the number of vertices, and the differences by monotonicity at lower orders. Every term is nonnegative. Integrating from independent spins then gives the sign rule as well.

Why this changes the paper

Monotonicity lets us bound a higher-order cumulant by products of lower-order ones. Iterating produces a tree bound with square roots of two-point correlations along its edges. Summing these trees controls the Taylor coefficients of $\log Z$ and gives a volume-uniform zero-free disk on the scale $m_\beta^{15/8}$.

The gap-comparison result is unchanged. The revision replaces the longer finite-box argument with this route and includes the component proof in Appendix A, rather than leaving monotonicity as a conditional input. The original research note describes the question and its earlier proof history.