Lower Bounds on Same-Set Inner Product in Correlated Spaces

Let P be a probability distribution over a finite alphabet Omega^L with all L marginals equal. Let X^(1), ..., X^(L), where X^(j) = (X_1^(j), ..., X_n^(j)) be random vectors such that for every coordinate i in [n] the tuples (X_i^(1), ..., X_i^(L)) are i.i.d. according to P.The question we address is: does there exist a function c_P independent of n such that for every f: Omega^n -u003e [0, 1] with E[f(X^(1))] = m u003e 0 we have E[f(X^(1)) * ... * f(X^(n))] u003e c_P(m) u003e 0?We settle the question for L=2 and when Lu003e2 and P has bounded correlation smaller than 1.