Abstract
For a sequence of Boolean functions \({f_n}:{\{ - 1,1\} ^{{V_n}}} \to \{ - 1,1\} \), 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.
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Acknowledgments
We are grateful to Itai Benjamini for his inspiring question that started this project. Our work was supported by the ERC Consolidator Grant 772466 “NOISE”. During most of this work, PG was a PhD student at the Central European University, Budapest. Thanks to Christophe Garban and Balázs Szegedy for reading the thesis, to Ohad Klein for a comment and a reference, and to two fantastic referees for their many corrections and suggestions.
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Galicza, P., Pete, G. Sparse reconstruction in spin systems. I: iid spins. Isr. J. Math. (2024). https://doi.org/10.1007/s11856-024-2606-0
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DOI: https://doi.org/10.1007/s11856-024-2606-0