Sparse reconstruction in spin systems. I: iid spins

Abstract

AbstractFor a sequence of Boolean functions $${f_n}:{\{ - 1,1\} ^{{V_n}}} \to \{ - 1,1\} $$ f n : { − 1 , 1 } V n → { − 1 , 1 } , defined on increasing configuration spaces of random inputs, we say that there is sparse reconstruction if there is a sequence of subsets Un ⊆ Vn of the coordinates satisfying ∣Un∣ = o(∣Vn∣) such that knowing the coordinates in Un gives us a non-vanishing amount of information about the value of fn.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 L2 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.