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Noise sensitivity of Boolean functions

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A Boolean function is a function from the hypercube {-1,1}^n into {-1,1}. Basic examples include the dictatorship function DICT = x_1 and the parity function PAR = x_1...x_n. We consider the effect of perturbing x_1,...,x_n by some small random noise. Clearly one would expect the dictatorship function to remain unchanged and the parity function to be almost independent from before. Our main interest is in Boolean functions arising in the percolation model where the effect of a random noise is less obvious. We shall see that phenomenon of noise sensitivity is related to the energy spectrum of a function. Thus Fourier analysis gives us a powerful tool for studying the affects of random noise on a percolation configuration.

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