  Monotone lattice recurrence relations such as the Frenkel-Kontorova lattice,
arise in Hamiltonian lattice mechanics as models for fe?rromagnetism and as
discretization of elliptic PDEs. Mathematically, they are a multidimensional
counterpart of monotone twist maps. They often admit a variational structure,
so that the solutions are the stationary points of a formal action function.
Classical Aubry-Mather theory establishes the existence of a large collection
of solutions of any rotation vector. For irrational rotation vectors this is
the well-known Aubry-Mather set. It consists of global minimizers and it may
have gaps.
  In this paper, we study the gradient flow of the formal action function and
we prove that every Aubry-Mather set can be interpolated by a continuous
gradient-flow invariant family, the so-called "ghost circle". The existence of
ghost circles is first proved for rational rotation vectors and Morse action
functions. The main technical result is a compactness theorem for ghost
circles, based on a parabolic Harnack inequality for the gradient flow, which
implies the existence of ghost circles of arbitrary rotation vectors and for
arbitrary actions. As a consequence, we can give a simple proof of the fact
that when an Aubry-Mather set has a gap, then this gap must be parametrized by
minimizers, or contain a non-minimizing solution.
