Lines and Networks
2014, v.20, Issue 1, 81-106
The theory of random lines has a celebrated history, reaching back 300 years into the past to the work of Buffon, and forming a major part of the field of stochastic geometry. Recently it has found application in the derivation of surprising non-stochastic results concerning effective planar networks [D.J. Aldous and W.S. Kendall, Short-length routes in low-cost networks via Poisson line patterns. Adv. Appl. Probab., 2008, v.40, N1, 1-21]. The following is an account corresponding to three lectures on this material, including a very brief introduction to the relevant stochastic geometry, and also a description of some more recent work concerning flows in related networks [W.S. Kendall, Networks and Poisson line patterns: fluctuation asymptotics. Oberwolfach Rep., 2008, v.5, N4, 2670-2672; Geodesics and flows in a Poissonian city. Ann. Appl. Probab., 2011, v.21, N3, 801-842] as well as a rather curious random metric space.
Keywords: stochastic geometry,random lines,planar network,effectiveness,flow