On the noise sensitivity of monotone functions

It is known that for all monotone functions f : {0, 1}" --> {0, 1}, if x is an element of {0, 1}" is chosen uniformly at random and y is obtained from x by flipping each of the bits of x independently with probability epsilon = n(-alpha), then P[f(x) not equal f(y)] < cn(-alpha+1/2), for some c > 0. Previously, the best construction of monotone functions satisfying P[f(n)(x) not equal f(n)(y)] greater than or equal to delta, where 0 < delta < 1/2, requiredepsilon greater than or equal to c(delta)n(-alpha), where alpha = 1 - ln 2/ln 3 = 0.36907(...), and c(delta) > 0. We improve this result by achieving for every 0 < delta < 1/2, P[f(n)(x) not equal f(n)(y)] greater than or equal to delta, with: epsilon = c(delta)n(-alpha) for any alpha < 1/2, using the recursive majority function with arity k = k(alpha); epsilon = c(delta)n(-1/2)log'n for t = log(2)rootpi/2 =.3257(...), using an explicit recursive majority function with increasing arities; and epsilon = c(delta)n(-1/2), nonconstructively, following a probabilistic CNF construction due to Tala-grand. We also study the problem of achieving the best dependence on delta in the case that the noise rate e is at least a small constant; the results we obtain are tight to within logarithmic factors. (C) 2003 Wiley Periodicals, Inc.