  In this paper we introduce a topology under which the pair empirical measure
of a large class of random walks satisfies a strong Large Deviation principle.
The definition of the topology is inspired by the recent article by Mukherjee
and Varadhan [16]. This topology is natural for translation-invariant problems
such as the downward deviations of the volume of a Wiener sausage or simple
random walk, known as the Swiss cheese model [18]. We also adapt our result to
some rescaled random walks and provide a contraction principle to the single
empirical measure despite a lack of continuity from the projection map, using
the notion of diagonal tightness.
