Large Deviations for Subcritical Bootstrap Percolation on the Erdos-Rényi Graph.

We study atypical behavior in bootstrap percolation on the Erdos-Rényi random graph. Initially a set S is infected. Other vertices are infected once at least r of their neighbors become infected. Janson et al. (Ann Appl Probab 22(5):1989-2047, 2012) locates the critical size of S, above which it is likely that the infection will spread almost everywhere. Below this threshold, a central limit theorem is proved for the size of the eventually infected set. In this work, we calculate the rate function for the event that a small set S eventually infects an unexpected number of vertices, and identify the least-cost trajectory realizing such a large deviation.

On the tail of the branching random walk local time.

Consider a critical branching random walk on Z d , d ≥ 1 , started with a single particle at the origin, and let L(x) be the total number of particles that ever visit a vertex x. We study the tail of L(x) under suitable conditions on the offspring distribution. In particular, our results hold if the offspring distribution has an exponential moment.

Limit theorem for the Robin Hood game.

In its simplest form, the Robin Hood game is described by the following urn scheme: every day the Sheriff of Nottingham puts s balls in an urn. Then Robin chooses r (r < s) balls to remove from the urn. Robin's goal is to remove balls in such a way that none of them are left in the urn indefinitely. Let T n be the random time that is required for Robin to take out all s · n balls put in the urn during the first n days. Our main result is a limit theorem for T n if Robin selects the balls uniformly at random. Namely, we show that the random variable T n · n -s/r converges in law to a Fréchet distribution as n goes to infinity.

The half plane UIPT is recurrent.

We prove that the half plane version of the uniform infinite planar triangulation (UIPT) is recurrent. The key ingredients of the proof are a construction of a new full plane extension of the half plane UIPT, based on a natural decomposition of the half plane UIPT into independent layers, and an extension of previous methods for proving recurrence of weak local limits (while still using circle packings).

Wishbone identifies bifurcating developmental trajectories from single-cell data.

Recent single-cell analysis technologies offer an unprecedented opportunity to elucidate developmental pathways. Here we present Wishbone, an algorithm for positioning single cells along bifurcating developmental trajectories with high resolution. Wishbone uses multi-dimensional single-cell data, such as mass cytometry or RNA-Seq data, as input and orders cells according to their developmental progression, and it pinpoints bifurcation points by labeling each cell as pre-bifurcation or as one of two post-bifurcation cell fates. Using 30-channel mass cytometry data, we show that Wishbone accurately recovers the known stages of T-cell development in the mouse thymus, including the bifurcation point. We also apply the algorithm to mouse myeloid differentiation and demonstrate its generalization to additional lineages. A comparison of Wishbone to diffusion maps, SCUBA and Monocle shows that it outperforms these methods both in the accuracy of ordering cells and in the correct identification of branch points.