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 ϵ = n^−α, then P[f(x) ≠ f(y)] < cn^−α+1/2, for some c > 0. We also study the problem of achieving the best dependence on δ in the case that the noise rate ϵ is at least a small constant; the results we obtain are tight to within logarithmic factors.