On the noise sensitivity of monotone functions

It is known that for all monotone functions f : {0, 1}n → {0, 1}, if x ∈ {0, 1}n is chosen uniformly at random and y is obtained from x by flipping each of the bits of x independently with probability = n, then P[f(x) 6= f(y)] < cn, for some c > 0. Previously, the best construction of monotone functions satisfying P[fn(x) 6= fn(y)] ≥ δ, where 0 < δ < 1/2, required ≥ c(δ)n, where α = 1 − ln 2/ ln 3 = 0.36907. . . , and c(δ) > 0. We improve this result by achieving for every 0 < δ < 1/2, P[fn(x) 6= fn(y)] ≥ δ, with: • = c(δ)n for any α < 1/2, using the recursive majority function with arity k = k(α); • = c(δ)n logn for t = log2 √ π/2 = .3257. . . , using an explicit recursive majority function with increasing arities; and, • = c(δ)n, non-constructively, following a probabilistic CNF construction due to Talagrand. We also study the problem of achieving the best dependence on δ in the case that the noise rate is at least a small constant; the results we obtain are tight to within logarithmic factors.