How Do Read-Once Formulae Shrink?

Let f be a de Morgan read-once function of n variables. Let f_ε be the random restriction obtained by independently assigning to each variable of f, the value 0 with probability (1 -ε)/2, the value 1 with the same probability, and leaving it unassigned with probability ε. We show that f_ε depends, on the average, on only O(ε^αn + εn^1/α) variables, where [formula omitted]. This result is asymptotically the tightest possible. It improves a similar result obtained recently by Håstad, Razborov and Yao.