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 by pinpointing bifurcation points and 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.

A construction as a growth process for sampling of the uniform infinite planar triangulation (UIPT), defined in [AnS], is given. The construction is algorithmic in nature, and is an efficient method of sampling a portion of the UIPT.By analyzing the progress rate of the growth process we show that a.s. the UIPT has growth rate r 4 up to polylogarithmic factors, in accordance with heuristic results from the physics literature. Additionally, the boundary component of the ball of radius r separating it from infinity a.s. has growth rate r 2 up to polylogarithmic factors. It is also shown that the properly scaled size of a variant of the free triangulation of an m-gon (also defined in [AnS]) converges in distribution to an asymmetric stable random variable of type 1/2.

The existence of the weak limit as n → ∞ of the uniform measure on rooted triangulations of the sphere with n vertices is proved. Some properties of the limit are studied. In particular, the limit is a probability measure on random triangulations of the plane.

A sorting network is a shortest path from 12 · · · n to n · · · 21 in the Cayley graph of S n generated by nearest-neighbour swaps. We prove that for a uniform random sorting network, as n → ∞ the spacetime process of swaps converges to the product of semicircle law and Lebesgue measure. We conjecture that the trajectories of individual particles converge to random sine curves, while the permutation matrix at half-time converges to the projected surface measure of the 2-sphere. We prove that, in the limit, the trajectories are Hölder-1/2 continuous, while the support of the permutation matrix lies within a certain octagon. A key tool is a connection with random Young tableaux. τ s 1 τ s 2 · · · τ s N = ρ

We study Bernoulli percolations on random lattices of the half-plane obtained as local limit of uniform planar triangulations or quadrangulations. Using the characteristic spatial Markov property or peeling process [5] of these random lattices we prove a surprisingly simple universal formula for the critical threshold for bond and face percolations on these graphs. Our techniques also permit us to compute off-critical and critical exponents related to percolation clusters such as the volume and the perimeter. d loc (m, m ) = 1 + R) −1 ,