How Do Read-Once Formulae Shrink?

Let f be a de Morgan read-once function of n variables. Let f " be the random restriction obtained by independently assigning to each variable of f, the value 0 with probability (1 ? ")=2, the value 1 with the same probability, and leaving it unassigned with probability ". We show that f " depends, on the average, on only O(" n + "n 1==) variables, where = log p 5?1 2 ' 3:27. This result is asymptotically the tightest possible. It improves a similar result obtained recently by H astad, Razborov and Yao.