## The Annals of Applied Probability

### Monotone properties of random geometric graphs have sharp thresholds

#### Abstract

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≥2. 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.

#### Article information

Source
Ann. Appl. Probab. Volume 15, Number 4 (2005), 2535-2552.

Dates
First available in Project Euclid: 7 December 2005

http://projecteuclid.org/euclid.aoap/1133965771

Digital Object Identifier
doi:10.1214/105051605000000575

Mathematical Reviews number (MathSciNet)
MR2187303

Zentralblatt MATH identifier
05039567

#### Citation

Goel, Ashish; Rai, Sanatan; Krishnamachari, Bhaskar. Monotone properties of random geometric graphs have sharp thresholds. Ann. Appl. Probab. 15 (2005), no. 4, 2535--2552. doi:10.1214/105051605000000575. http://projecteuclid.org/euclid.aoap/1133965771.

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