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In many problems of solid mechanics (e.g., stochastic finite elements, statistical fracture mechanics) there is a need for resolution of dependent fields over scales not infinitely larger than the microscale. This task may be accomplished through a "mesa-scale window" which becomes the classical Representative Volume Element (RVE) in the infinite limit relative to the microscale. It turns out that the material properties at such a mesoscale cannot be uniquely approximated by a random field of stiffness/compliance with locally isotropic realizations, but, rather, two random continuum fields with locally anisotropic realizations, corresponding respectively to Dirichlet and Neumann boundary conditions on the mesa-scale, need to be introduced to bound the material response from above and from below. We discuss statistical characteristics of these two mesoscale random fields, including their spatial correlation structure, for anti-plane elastic response of random two-phase composites with Voronoi geometry at the percolation point. Particular attention is given to the scaling of effective responses obtained from both conditions, which sheds light on the minimum acceptable size of an RVE.