, Volume 77, Issue 1-2, pp 55-64

The influence of variables in product spaces

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Abstract

LetX be a probability space and letf: X n → {0, 1} be a measurable map. Define the influence of thek-th variable onf, denoted byI f (k), as follows: Foru=(u 1,u 2,…,u n−1) ∈X n−1 consider the setl k (u)={(u 1,u 2,...,u k−1,t,u k ,…,u n−1):tX}. \(I_f (k) = \Pr (u \in X^{n - 1} :f is not constant on l_k (u)).\)

More generally, forS a subset of [n]={1,...,n} let the influence ofS onf, denoted byI f (S), be the probability that assigning values to the variables not inS at random, the value off is undetermined.

Theorem 1:There is an absolute constant c 1 so that for every function f: X n → {0, 1},with Pr(f −1(1))=p≤1/2,there is a variable k so that \(I_f (k) \geqslant c_1 p\frac{{\log n}}{n}.\)

Theorem 2:For every f: X n → {0, 1},with Prob(f=1)=1/2, and every ε>0,there is S ⊂ [n], |S|=c 2(ε)n/logn so that I f (S)≥1−ε.

These extend previous results by Kahn, Kalai and Linial for Boolean functions, i.e., the caseX={0, 1}.

Work supported in part by grants from the Binational Israel-US Science Foundation and the Israeli Academy of Science.