"The new world of infinite random geometric graphs" presented by Anthony Bonato (Ryerson)
Math & Stat Department, Colloquium Room #319, Chase Building, Dalhousie University
The infinite random or Rado graph R has been of interest to graph theorists, probabilists, and logicians for the last half-century. The graph R has many peculiar properties, such as its categoricity : R is the unique countable graph satisfying certain adjacency properties. Erd ̋os and R ́enyi proved in 1963 that a countably infinite binomial random graph is isomorphic to R.
Random graph processes giving unique limits are, however, rare. Recent joint work with Jeannette Janssen proved the existence of a family of random geometric graphs with unique limits. These graphs arise in the normed space`n∞, which consists of Rn equipped with the L∞-norm. Balister, Bollob ́as, Gunderson, Leader, and Walters used tools from functional analysis to show that these unique limit graphs are deeply tied to the L ∞ -norm. Precisely, a random geometric graph on any normed, finite-dimensional space not isometric ` n ∞ gives non-isomorphic limits with probability 1.
With Janssen and Anthony Quas, we have discovered unique limits in infinite di-mensional settings including sequences spaces and spaces of continuous functions. We survey these newly discovered infinite random geometric graphs and their properties.