reduction is expected to hold and so this theory is generally thought to be completely independent of its volume. In the current paper we analyze this volume dependence. Our motivation is two-fold. Firstly, the ‘t Hooft model is analytically soluble at large-$N$. Thus we can explicitly see how large-$N$ volume reduction works in this case, and what may cause it to fail. This topic was also addressed for zero baryon number by the authors of Ref. [@SchonThies_decompact], and our treatment here differs from that paper by being manifestly gauge invariant, by going beyond zero baryon number, and by using the lattice regularization. Our approach also makes a direct connection with Eguchi-Kawai reduction, and shows how the expectation values of spatial Polyakov loops play a crucial role in the validity of volume independence.

Secondly, this paper is a prelude to our companion publication Ref. [@nonzeroBpaper] where we use the formalism presented here to solve the theory in the presence of nonzero baryon density. Considering the current incomplete understanding of the way four-dimensional QCD behaves at low temperatures and large (but not asymptotic) baryon densities, we believe that such a study is useful. Also, there exist certain confusions in the literature about 