Direct products in communication complexity

We give exponentially small upper bounds on the success probability for computing the direct product of any function over any distribution using a communication protocol. Let suc(μ, f,C) denote the maximum success probability of a 2-party communication protocol for computing the boolean function f(x, y) with C bits of communication, when the inputs (x, y) are drawn from the distribution μ. Let μn be the product distribution on n inputs and fn denote the function that computes n copies of f on these inputs. We prove that if T log^3/2T ⋘ (C - 1) √ n and suc(μ, f,C) < 2 3 , then suc(μn, fⁿ, T) ≤ exp(-Ω(n)). When μ is a product distribution, we prove a nearly optimal result: As long as T log ² T ⋘ Cn, we must have suc(μn, fn, T) ≤ exp(-Ω(n)).