  We propose to compute physical properties by Monte Carlo calculations using
conditional expectation values. The latter are obtained on top of the usual
Monte Carlo sampling by partitioning the physical space in several subspaces or
fragments, and subsampling each fragment (i.e., performing side walks) while
freezing the environment. No bias is introduced and a zero-variance principle
holds in the limit of separability, i.e. when the fragments are independent. In
practice, the usual bottleneck of Monte Carlo calculations -- the scaling of
the statistical fluctuations as a function of the number of particles N -- is
relieved for extensive observables. We illustrate the method in variational
Monte Carlo on the 2D Hubbard model and on metallic hydrogen chains using
Jastrow-Slater wave functions. A factor O(N) is gained in numerical efficiency.
