Average Degree in the Interval Graph of a Random Boolean Function

We consider an n-ary random Boolean function f such that Pr[f(α̃) = 1] = p for α̃ ∈ {0, 1} and study its geometric model, the so called interval graph. The interval graph of a Boolean function was introduced by Sapozhenko and has been used in construction of schemes realizing Boolean functions. Using this model, we estimate the number of maximal intervals intersecting a given maximal interval of a random Boolean function and prove that the asymptotic bound on the logarithm of the number is (1 + φ(n)) log2 log1/p n · log2 n, where φ(n) → 0 as n → ∞.