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For the exactly solvable model of exponential last passage percolation on $\mathbb{Z}^2$, it is known that given any non-axial direction, all the semi-infinite geodesics starting from points in $\mathbb{Z}^2$ in that direction almost surely…
For first passage percolation (FPP) on integer lattice with i.i.d. passage time distributions, in order to show existence of semi-infinite geodesics along a fixed direction, one requires unproven assumptions on the limiting shape. We…
We study the problem of estimating the barycenter of a distribution given i.i.d. data in a geodesic space. Assuming an upper curvature bound in Alexandrov's sense and a support condition ensuring the strong geodesic convexity of the…
By measuring or calculating coalescence times for several models of coalescence or evolution, with and without selection, we show that the ratios of these coalescence times become universal in the large size limit and we identify a few…
We consider planar directed last-passage percolation on the square lattice with general i.i.d. weights and study the geometry of the full set of semi-infinite geodesics in a typical realization of the random environment. The structure of…
A well-known question in planar first-passage percolation concerns the convergence of the empirical distribution of weights as seen along geodesics. We demonstrate this convergence for an explicit model, directed last-passage percolation on…
The coefficients in a second order parabolic linear stochastic partial differential equation (SPDE) are estimated from multiple spatially localised measurements. Assuming that the spatial resolution tends to zero and the number of…
For any graph having a suitable uniform Poincare inequality and volume growth regularity, we establish two-sided Gaussian transition density estimates and parabolic Harnack inequality, for constant speed continuous time random walks…
The characteristic time for the electrocoalescence of two sessile drops is identified and verified for wide range of operating parameters such as Ohnesorge, Electrowetting number, driving frequency and drop to surrounding medium viscosity.…
We consider a wide class of ergodic first passage percolation processes on Z^2 and prove that there exist at least four one-sided geodesics a.s. We also show that coexistence is possible with positive probability in a four color…
We consider species tree estimation under a standard stochastic model of gene tree evolution that incorporates incomplete lineage sorting (as modeled by a coalescent process) and gene duplication and loss (as modeled by a branching…
In last passage percolation models, the energy of a path is maximized over all directed paths with given endpoints in a random environment, and the maximizing paths are called geodesics. The geodesics and their energy can be scaled so that…
Random growth models are fundamental objects in modern probability theory, have given rise to new mathematics, and have numerous applications, including tumor growth and fluid flow in porous media. In this article, we introduce some of the…
A popular line of research in evolutionary biology is the use of time-calibrated phylogenies for the inference of diversification processes. This requires computing the likelihood of a given ultrametric tree as the reconstructed tree…
For exactly solvable models of planar last passage percolation, it is known that geodesics of length $n$ exhibit transversal fluctuations at scale $n^{2/3}$ and matching (up to exponents) upper and lower bounds for the tail probabilities…
We consider the geodesic of the directed last passage percolation with iid exponential weights. We find the explicit one-point distribution of the geodesic location joint with the last passage times, and its limit as the parameters go to…
We consider geodesics for first passage percolation (FPP) on $\mathbb{Z}^d$ with iid passage times. As has been common in the literature, we assume that the FPP system satisfies certain basic properties conjectured to be true, and derive…
The problem of estimating the growth rate of a birth and death processes based on the coalescence times of a sample of $n$ individuals has been considered by several authors (\cite{stadler2009incomplete, williams2022life,…
The problem of estimating the probability of a random process reaching a certain level is well known. In this article, two-sided estimates are established for the probability that a regenerative process reaches a high level. Two auxiliary…
Species tree reconstruction is complicated by effects of Incomplete Lineage Sorting (ILS), commonly modeled by the multi-species coalescent model. While there has been substantial progress in developing methods that estimate a species tree…