Title: Rigidity percolation in random frameworks, periodic framework, and applications
Abstract: A framework is a structure made of fixed-length bars connected by universal joints with full rotational freedom. The allowed motions of the joints are those that preserve the lengths and connectivity of the bars. If all the allowed continuous motions are Euclidean isometries, then the framework is said to be rigid, and otherwise it is flexible.
Although both the model (frameworks) and the question (rigidity versus flexibility) are quite simple, both arise naturally in a number of applications, including crystallography, condensed matter physics, protein biology and sensor networks.
I’ll give an introduction to the main ideas in the field and then talk about two of my recent results:
(1) A “sharp threshold” for a large rigid substructure to emerge in a random framework in the plane. This is a finite analogue of “rigidity percolation” that had been studied in simulation by physicists as a model for phase transitions in glasses, and this result confirms theoretically their observations. (Joint with Shiva Kasiviswanathan and Cris Moore)
(2) A new combinatorial theory for rigidity and flexibility in “generic planar periodic frameworks”. These are infinite frameworks, periodic with respect to a lattice, that arise in crystallographic applications. I’ll describe an efficiently checkable, combinatorial characterization of rigidity and flexibility for this class. (Joint with Justin Malestein)