Sensitivity, Stability, and Computation

ended

Noise sensitivity and noise stability of Boolean functions, percolation, and other models were introduced in a paper by Benjamini, Kalai, and Schramm (1999) and were extensively studied in the last two decades. We propose to extend this study to various stochastic and combinatorial models, and to explore connections with computer science, quantum information, voting methods and other areas. The first goal of our proposed project is to push the mathematical theory of noise stability and noise sensitivity forward for various models in probabilistic combinatorics and statistical physics. A main mathematical tool, going back to Kahn, Kalai, and Linial (1988), is applications of (high-dimensional) Fourier methods, and our second goal is to extend and develop these discrete Fourier methods. Our third goal is to find applications toward central old-standing problems in combinatorics, probability and the theory of computing. The fourth goal of our project is to further develop the ``argument against quantum computers'' which is based on the insight that noisy intermediate scale quantum computing is noise stable. This follows the work of Kalai and Kindler (2014) for the case of noisy non-interacting bosons. The fifth goal of our proposal is to enrich our mathematical understanding and to apply it, by studying connections of the theory with various areas of theoretical computer science, and with the theory of social choice.