Monte Carlo integration is more efficient in higher dimensions. To see this, let us assume that our integration volume is a hypercube with side \( L \) and dimension \( d \). This cube contains hence \( N=(L/h)^d \) points and therefore the error in the result scales as \( N^{-k/d} \) for the traditional methods.

The error in the Monte carlo integration is however independent of \( d \) and scales as $$ \mathrm{error}\sim 1/\sqrt{N}. $$ Always!

Comparing this error with that of the traditional methods, shows that Monte Carlo integration is more efficient than an algorithm with error in powers of \( k \) when $$ d>2k. $$