Minimum spanning trees across dense cities

Consider~$n$ nodes distributed independently across~$N$ cities contained with the unit square~$S$ according to a distribution~$f.$ Each city is modelled as an~$r_n \times r_n$ square contained within~$S$ and~$MSTC_n$ denotes the length of the minimum spanning tree containing all the~$n$ nodes. We use approximation methods to obtain variance estimates for~$MSTC_n$ and prove that if the cities are well-connected and densely populated in a certain sense, then~$MSTC_n$ appropriately centred and scaled converges to zero in probability. Using the proof techniques, we alternately derive corresponding results for the length~$MST_n$ of the minimum spanning tree for the usual case when the nodes are independently distributed throughout the unit square~$S.$ In particular, we obtain that the variance of~$MST_n$ grows at most as a power of the logarithm of~$n$ and use a subsequence argument to get almost sure convergence of~$MST_n$ appropriately centred and scaled.