Random walks on the circle and Diophantine approximation.

Random walks on the circle group whose elementary steps are lattice variables with span or taken mod exhibit delicate behavior. In the rational case, we have a random walk on the finite cyclic subgroup , and the central limit theorem and the law of the iterated logarithm follow from classical results on finite state space Markov chains. In this paper, we extend these results to random walks with irrational span , and explicitly describe the transition of these Markov chains from finite to general state space as along the sequence of best rational approximations.

An automated, data-driven approach to children's social dynamics in space and time.

Most children first enter social groups of peers in preschool. In this context, children use movement as a social tool, resulting in distinctive proximity patterns in space and synchrony with others over time. However, the social implications of children's movements with peers in space and time are difficult to determine due to the difficulty of acquiring reliable data during natural interactions.

Locally common graphs.

Goodman proved that the sum of the number of triangles in a graph on nodes and its complement is at least ; in other words, this sum is minimized, asymptotically, by a random graph with edge density 1/2. Erdős conjectured that a similar inequality will hold for in place of , but this was disproved by Thomason. But an analogous statement does hold for some other graphs, which are called .