Sparse reconstruction in spin systems II: Ising and other factor of IID measures

TL;DR

This paper investigates the possibility of sparse reconstruction in spin systems, showing it is generally impossible in high-temperature regimes but possible in low-temperature regimes for certain models, using diverse mathematical methods.

Contribution

It extends previous work by analyzing sparse reconstruction in spin systems related to IID measures, providing new results for the Ising model across different temperature regimes.

Findings

No sparse reconstruction in high-temperature regimes for Ising models on Euclidean boxes and Curie-Weiss.

Sparse reconstruction is possible in critical and low-temperature regimes for the majority function.

Provides quantitative bounds for two-dimensional boxes and the Curie-Weiss model.

Abstract

For a sequence of Boolean functions $f_n : \{-1, 1\}^{V_n} \longrightarrow \{-1, 1\}$, with random input given by some probability measure $\mathbb{P}_n$, we say that there is sparse reconstruction for $f_n$ if there is a sequence of subsets $U_n \subseteq V_n$ of coordinates satisfying $|U_n| = o(|V_n|)$ such that knowing the spins in $U_n$ gives us a non-vanishing amount of information about the value of $f_n$. In the first part of this work, we showed that if the $\mathbb{P}_n$s are product measures, then no sparse reconstruction is possible for any sequence of transitive functions. In this sequel, we consider spin systems that are relatives of IID measures in one way or another, with our main focus being on the Ising model on finite transitive graphs or exhaustions of lattices. We prove that no sparse reconstruction is possible for the entire high temperature regime on Euclidean boxes and the Curie-Weiss model, while sparse reconstruction for the majority function of the spins is possible in the critical and low temperature regimes. We give quantitative bounds for two-dimensional boxes and the Curie-Weiss model, sharp in the latter case. The proofs employ several different methods, including factor of IID and FK random cluster representations, strong spatial mixing, a generalization of discrete Fourier analysis to Divide-and-Color models, and entropy inequalities.