One spin per correlation volume: a conditional Simon inequality and the cost of reading a ferromagnet

Abstract

How many spins of a ferromagnet must one observe to learn its magnetization? For the ferromagnetic Ising model in zero field on a finite graph we prove that, for every set U of observed sites and all sites x, y, E[ E[σ_x | σ_U] E[σ_y | σ_U] ] ≤ Σ_{u∈U} ⟨σ_x σ_u⟩⟨σ_u σ_y⟩. This extends Simon's correlation inequality from separating sets to arbitrary sets. The proof combines the Ding–Song–Sun inequality with the FKG inequality. It follows that observed spins never explain more of a positive linear statistic jointly than one at a time, and that on a vertex-transitive graph with N sites and susceptibility χ, any b observed spins explain at most a fraction bχ/N of the variance of the magnetization, whichever spins are chosen and however they are combined. Since O(N/χ) randomly placed spins suffice, reading the magnetization costs one spin per correlation volume. For the critical Ising model on the n × n torus this gives clue(M_n | U) ≤ C|U| n^(-1/4) for every set U. Galicza and Pete showed that suitably chosen sets of size ≫ n^(1/4) suffice, so n^(1/4) is the sparse-reconstruction threshold for the magnetization; the same holds on boxes with free or plus boundary conditions. For the majority function the exact inequality fails, and its threshold remains open. We extend the bound, up to a constant factor, to continuous-spin ferromagnets in the Ellis–Monroe–Newman class, including φ⁴, and exactly to one step of synchronous heat-bath dynamics. For the diffusion that arises as the scaling limit of the critical Curie–Weiss model we prove the corresponding bound for predicting the future, Var(E[Y_τ | √c Y_0 + Z]) ≤ c (E[Y_0 Y_τ])². Read in Curie–Weiss terms, it says that b present spins explain at most a fraction (bχ/N)ρ(t)² of the variance of the magnetization at time t, where ρ is its autocorrelation; the passage from finite N to the limit is not proved here. For Glauber dynamics on general graphs the analogous bound held in every exact computation we made, but the natural proof fails and the question remains open. Files: the paper (PDF), its LaTeX source, and Python code that reproduces every computational check in the paper. This preprint was developed with substantial AI assistance (Claude, Anthropic) and has not yet been peer reviewed; see the statement at the end of the paper.