Asymptotic enumeration of binary contingency tables and comparison with independence heuristic

For parameters $n,\delta,B,C$, we obtain sharp asymptotic formula for number of $(n+\lfloor n^\delta\rfloor)^2$ dimensional binary contingency tables with non-uniform margins $\lfloor BCn\rfloor$ and $\lfloor Cn\rfloor$. Furthermore, we compare our results with the classical \textit{independent heuristic} and prove that the independent heuristic overestimates by a factor of $e^{\Theta(n^{2\delta})}$. Our comparison is based on the analysis of the \textit{correlation ratio} and we obtain the explicit bound for the constant in $\Theta$.