Traveling salesman problem across dense cities

Consider~$n$ nodes~$\{X_i\}_{1 \leq i \leq n}$ 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 let~$TSPC_n$ denote the length of the minimum length cycle containing all the~$n$ nodes, corresponding to the traveling salesman problem (TSP). We obtain variance estimates for~$TSPC_n$ and prove that if the cities are well-connected and densely populated in a certain sense, then~$TSPC_n$ appropriately centred and scaled converges to zero in probability. We also obtain large deviation type estimates for~$TSPC_n.$ Using the proof techniques, we alternately obtain corresponding results for the length~$TSP_n$ of the minimum length cycle in the unconstrained case, when the nodes are independently distributed throughout the unit square~$S.$