Sparse reconstruction in spin systems I: iid spins

TL;DR

This paper investigates the possibility of sparse reconstruction in spin systems, showing that for certain classes of functions and models, such reconstruction is impossible, with implications for understanding information flow in these systems.

Contribution

It introduces conditions under which sparse reconstruction cannot occur in spin systems, extending results to non-transitive functions and critical percolation models.

Findings

No sparse reconstruction for transitive functions under product measures.

Sparse reconstruction is also impossible for critical planar percolation crossing events.

Connections established with information theory and cooperative game theory.

Abstract

For a sequence of Boolean functions $f_n : \{-1,1\}^{V_n} \longrightarrow \{-1,1\}$, defined on increasing configuration spaces of random inputs, we say that there is sparse reconstruction if there is a sequence of subsets $U_n \subseteq V_n$ of the coordinates satisfying $|U_n| = o(|V_n|)$ such that knowing the coordinates in $U_n$ gives us a non-vanishing amount of information about the value of $f_n$. We first show that, if the underlying measure is a product measure, then no sparse reconstruction is possible for any sequence of transitive functions. We discuss the question in different frameworks, measuring information content in $L^2$ and with entropy. We also highlight some interesting connections with cooperative game theory. Beyond transitive functions, we show that the left-right crossing event for critical planar percolation on the square lattice does not admit sparse reconstruction either. Some of these results answer questions posed by Itai Benjamini.