  We prove a quenched functional central limit theorem (quenched FCLT) for the
sums of a random field (r.f.) along a Z d-random walk in different frameworks:
probabilistic (when the r.f. is i.i.d. or a moving average of i.i.d. random
variables) and algebraic (when the r.f. is generated by commuting automorphisms
of a torus or by commuting hyperbolic flows on homogeneous spaces).
