A simulation of an interactive protocol entails the use of interactive communication to produce the output of the protocol to within a fixed statistical distance $\epsilon$. Recent works have proposed that the information complexity of the protocol plays a central role in characterizing the minimum number of bits that the parties must exchange for a successful simulation, namely the distributional communication complexity of simulating the protocol. Several simulation protocols have been proposed with communication complexity depending on the information complexity of the simulated protocol. However, in the absence of any general lower bounds for distributional communication complexity, the conjectured central role of information complexity is far from settled. We fill this gap and show that the distributional communication complexity of $\epsilon$-simulating a protocol is bounded below by the $\epsilon$-tail $\lambda_\epsilon$ of the information complexity density, a random variable with information complexity as its expected value. For protocols with bounded number of rounds, we give a simulation protocol that yields a matching upper bound. Thus, it is not information complexity but $\lambda_\epsilon$ that governs the distributional communication complexity.

As applications of our bounds, in the amortized regime for product protocols, we identify the exact second order term, together with the precise dependence on $\epsilon$. For general protocols such as a mixture of two product protocols or for the amortized case when the repetitions are not independent, we derive a general formula for the leading asymptotic term. These results sharpen and significantly extend known results in the amortized regime. In the single-shot regime, our lower bound clarifies the dependence of communication complexity on $\epsilon$. We illustrate this with an example that exhibits an arbitrary separation between distributional communication complexity and information complexity for all sufficiently small $\epsilon$.

A simulation of an interactive protocol entails the use of an interactive communication to produce the output of the protocol to within a fixed statistical distance $\epsilon$. Recent works in the TCS community have propagated that the information complexity of the protocol plays a central role in characterizing the minimum number of bits that the parties must exchange for a successful simulation, namely the distributional communication complexity of simulating the protocol. Several simulation protocols have been proposed with communication complexity depending on the information complexity of the simulated protocol. However, in the absence of any general lower bounds for distributional communication complexity, the conjectured central role of information complexity is far from settled. We fill this gap and show that the distributional communication complexity of $\epsilon$-simulating a protocol is bounded below by the $\epsilon$-tail $\lambda_\epsilon$ of the information complexity density, a random variable with information complexity as its expected value. For protocols with bounded number of rounds, we give a simulation protocol that yields a matching upper bound. Thus, it is not information complexity but $\lambda_\epsilon$ that governs the distributional communication complexity.

As applications of our bounds, in the amortized regime for product protocols, we identify the exact second order term, together with the precise dependence on $\epsilon$. For general protocols such as a mixture of two product protocols or for the amortized case when the repetitions are not independent, we derive a general formula for the leading asymptotic term. These results sharpen and significantly extend known results in the amortized regime. In the single-shot regime, our lower bound clarifies the dependence of communication complexity on $\epsilon$. We illustrate this with an example that exhibits an arbitrary separation between distributional communication complexity and information complexity for all sufficiently small $\epsilon$.