Monotone properties of random geometric graphs have sharp thresholds

Random geometric graphs result from taking $n$ uniformly distributed points in the unit cube, $[0,1]^d$, and connecting two points if their Euclidean distance is at most $r$, for some prescribed $r$. We show that monotone properties for this class of graphs have sharp thresholds by reducing the problem to bounding the bottleneck matching on two sets of $n$ points distributed uniformly in $[0,1]^d$. We present upper bounds on the threshold width, and show that our bound is sharp for $d=1$ and at most a sublogarithmic factor away for $d\ge2$. Interestingly, the threshold width is much sharper for random geometric graphs than for Bernoulli random graphs. Further, a random geometric graph is shown to be a subgraph, with high probability, of another independently drawn random geometric graph with a slightly larger radius; this property is shown to have no analogue for Bernoulli random graphs.Comment: Published at http://dx.doi.org/10.1214/105051605000000575 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

On the optimal map in the 2-dimensional random matching problem

We show that, on a $2$-dimensional compact manifold, the optimal transport map in the semi-discrete random matching problem is well-approximated in the $L^2$-norm by identity plus the gradient of the solution to the Poisson problem $-\Delta f^{n,t} = \mu^{n,t}-1$, where $\mu^{n,t}$ is an appropriate regularization of the empirical measure associated to the random points. This shows that the ansatz of Caracciolo et al. (Scaling hypothesis for the Euclidean bipartite matching problem) is strong enough to capture the behavior of the optimal map in addition to the value of the optimal matching cost. As part of our strategy, we prove a new stability result for the optimal transport map on a compact manifold.Comment: 20 page

Spectral Analysis of the Adjacency Matrix of Random Geometric Graphs

International audienceIn this article, we analyze the limiting eigen-value distribution (LED) of random geometric graphs (RGGs). The RGG is constructed by uniformly distributing n nodes on the d-dimensional torus T and connecting two nodes if their p-distance, p in [1, ∞] is at most r. In particular, we study the LED of the adjacency matrix of RGGs in the connectivity regime, in which the average vertex degree scales as log(n) or faster, i.e., Ω (log(n)). In the connectivity regime and under some conditions on the radius r, we show that the LED of the adjacency matrix of RGGs converges to the LED of the adjacency matrix of a deterministic geometric graph (DGG) with nodes in a grid as n goes to infinity. Then, for n finite, we use the structure of the DGG to approximate the eigenvalues of the adjacency matrix of the RGG and provide an upper bound for the approximation error