Over time, a population acquires neutral genetic substitutions as a consequence of random drift. A famous result in population genetics asserts that the rate, K, at which these substitutions accumulate in the population coincides with the mutation rate, u, at which they arise in individuals: K = u. This identity enables genetic sequence data to be used as a “molecular clock” to estimate the timing of evolutionary events. While the molecular clock is known to be perturbed by selection, it is thought that K = u holds very generally for neutral evolution. Here we show that asymmetric spatial population structure can alter the molecular clock rate for neutral mutations, leading to either K<u or K>u. Our results apply to a general class of haploid, asexually reproducing, spatially structured populations. Deviations from K = u occur because mutations arise unequally at different sites and have different probabilities of fixation depending on where they arise. If birth rates are uniform across sites, then K
u. In general, K can take any value between 0 and Nu. Our model can be applied to a variety of population structures. In one example, we investigate the accumulation of genetic mutations in the small intestine. In another application, we analyze over 900 Twitter networks to study the effect of network topology on the fixation of neutral innovations in social evolution.

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We consider two-dimensional critical bond percolation. Conditioned on the existence of an open circuit in an annulus, we show that the ratio of the expected size of the shortest open circuit to the expected size of the innermost circuit tends to zero as the side length of the annulus tends to infinity, the aspect ratio remaining fixed. The same proof yields a similar result for the lowest open crossing of a rectangle. In this last case, we answer a question of Kesten and Zhang by showing in addition that the ratio of the length of the shortest crossing to the length of the lowest tends to zero in probability. This suggests that the chemical distance in critical percolation is given by an exponent strictly smaller than that of the lowest path.

We study novel variations of Voronoi games (and associated random processes) that we call Voronoi choice games. These games provide a rich framework for studying questions regarding the power of small numbers of choices in multi-player, competitive scenarios in a geometric setting. Our work can be seen as a natural variation of the classical framework of Hotelling. While games based on Voronoi diagrams have appeared recently in the literature, our variations appear to be the first that take into account a notion of limited choices and the corresponding asymmetries in player strategies. As a problem example, suppose a group of n miners (or players) are staking land claims through the following process: each miner has m associated points in the unit square (or, for symmetry, the unit torus), so the kth miner will have associated points pk1, pk2, . . . , pkm. We generally here think of m as being a small constant, such as 2. Each miner chooses one of these points as the base point for their claim. Generally the points will be distinct for each miner. Each miner obtains mining rights for the area of the square that is closest to their chosen base; that is, they obtain the Voronoi cell corresponding to their chosen point in the Voronoi diagram of the n chosen points. Each player’s goal is simply to maximize the amount of land under their control. What can we say about the players’ strategy and the equilibria of such games? We provide several results on games in this setting. For example, we show that a correlated equilibrium can be found in polynomial time for the natural 1, 2, and 3-dimensional variations of the problem. For the one-dimensional version of the problem on the unit circle, we show that a pure Nash equilibrium exists when each player owns the part of the circle nearest to their point, but it is NP-hard to determine whether a pure Nash equilibrium exists when each player owns the arc starting from their point clockwise to the next point. We also consider a random version of the game, where players have m points chosen uniformly at random. We derive bounds on the expected number of pure Nash equilibria for a variation of the 1-dimensional game in this setting using interesting properties of random arc lengths on circles.

In this paper, we study the inviscid limit of the free surface incompressible Navier-Stokes equations with or without surface tension. By delicate estimates, we prove the weak boundary layer of the velocity of the free surface Navier-Stokes equations and the existence of strong or weak vorticity layer for different conditions. When the limit of the difference between the initial Navier-Stokes vorticity and the initial Euler vorticity is nonzero, or the tangential projection on the free surface of the Euler strain tensor multiplying by normal vector is nonzero, there exists a strong vorticity layer. Otherwise, the vorticity layer is weak. We estimate convergence rates of tangential derivatives and the first order standard normal derivative in energy norms, we show that not only tangential derivatives and standard normal derivative have different convergence rates, but also their convergence rates are different for different Euler boundary data. Moreover, we determine regularity structure of the free surface Navier-Stokes solutions with or without surface tension, surface tension changes regularity structure of the solutions.