Given a 'cost' functional F on paths γ in a domain D ⊂ R d , in the form F (γ) = 1 0 f (γ(t),γ(t))dt, it is of interest to approximate its minimum cost and geodesic paths. Let X 1 , . . . , Xn be points drawn independently from D according to a distribution with a density. Form a random geometric graph on the points where X i and X j are connected when 0 < |X i − X j | < , and the length scale = n vanishes at a suitable rate.For a general class of functionals F , associated to Finsler and other distances on D, using a probabilistic form of Gamma convergence, we show that the minimum costs and geodesic paths, with respect to types of approximating discrete 'cost' functionals, built from the random geometric graph, converge almost surely in various senses to those corresponding to the continuum cost F , as the number of sample points diverges. In particular, the geodesic path convergence shown appears to be among the first results of its kind.2010 Mathematics Subject Classification. 60D05, 58E10, 62-07, 49J55, 49J45, 53C22, 05C82.

We consider a large class of random geometric graphs constructed from samples Xn = {X 1 , X 2 , . . . , Xn} of independent, identically distributed observations of an underlying probability measure ν on a bounded domain D ⊂ R d . The popular 'modularity' clustering method specifies a partition Un of the set Xn as the solution of an optimization problem. In this paper, under conditions on ν and D, we derive scaling limits of the modularity clustering on random geometric graphs. Among other results, we show a geometric form of consistency: When the number of clusters is a priori bounded above, the discrete optimal partitions Un converge in a certain sense to a continuum partition U of the underlying domain D, characterized as the solution of a type of Kelvin's shape optimization problem.2010 Mathematics Subject Classification. 60D05,62G20,05C82,49J55,49J45,68R10.