We study monotonicity testing of Boolean functions over the hypergrid [n]^d and design a non-adaptive tester with 1-sided error whose query complexity is O(d^{5/6}) \poly(\log n,1/\eps). Previous to our work, the best known testers had query complexity linear in d but independent of n. We improve upon these testers as long as n = 2^{d^{o(1)}}.
To obtain our results, we work with what we call "the augmented hypergrid", which adds extra edges to the hypergrid. Our main technical contribution is a Margulis-style isoperimetric result for the augmented hypergrid, and our tester, like previous testers for the hypercube domain, performs directed random walks on this structure.