The bullet problem with discrete speeds

Bullets are fired from the origin of the positive real line, one per second, with independent speeds sampled uniformly from a discrete set. Collisions result in mutual annihilation. We show that a bullet with the second largest speed survives with positive probability, while a bullet with the smallest speed does not. This also holds for exponential spacings between firing times. Our results imply that the middle-velocity particle survives with positive probability in a two-sided version of the bullet process with three speeds known to physicists as ballistic annihiliation

Frog Model Wakeup Time on the Complete Graph

The frog model is a system of random walks where active particles set sleeping particles in motion. On the complete graph with n vertices it is equivalent to a well-understood rumor spreading model. We given an alternate and elementary proof that the wakeup time, that is, the expected time for every particle to be activated, is &Theta(log n). Additionally, we give an explicit distributional equation for the wakeup time as a mixture of geometric random variables

Interaction between Convection and SST in Tropical RCE

Thesis (Master's)--University of Washington, 2020Tropical convection is a significant source of uncertainty in predicting climate sensitivity with models. Radiative convective equilibrium (RCE), in which radiative cooling is balanced by convective heating, is currently believed to be a useful approximation of the tropics. Using this simplified model, we study the interactions between tropical convection and sea surface temperature (SST) gradients. Our model configuration is an aquaplanet with uniform solar irradiance, no rotation and interactive SST. Experiments run in this configuration show a spontaneous clumping of convection (self aggregation) that divides the domain into two regions: an area of upward motion over warm SSTs and an area of downward motion over cold SSTs. The system does not stay aggregated, however, and cycles between aggregated and disaggregated states on time scales longer than a year. To better understand the interplay between SST gradients and convection, we focus on the temperature contrast between the warm and cold pools of SST. By analyzing the main mechanisms that drive and limit this cycle, and how these mechanisms change with warming, we hope to better understand limits on SST variability in the real tropics. The main mechanisms responsible for driving the temperature contrast cycle, in this particular model, are gradients in surface cooling by evaporation and shortwave shading by clouds. In the region of upward motion, the shortwave shading is dominated by the ice water path from increased convection at temperatures below freezing. In the region of subsiding motion, low clouds dominate. Similarly, the evaporation in the region of upward motion is controlled almost exclusively by the surface wind speed, while the evaporation in the subsiding region is driven approximately equally by surface winds and saturation deficit. Beyond using the cycle to analyze how the SST contrast is maintained, we turn to the mean state to answer the question of how the SST contrast will change with warming. Regardless of warming by increasing insolation or CO_2, the SST contrast initially increases, and then decreases for the hottest cases. This decrease for the hottest cases appears to be associated with the beginning of the runaway Greenhouse Effect. For the cases most like the current tropics, however, the SST contrast increases with warming. This is not due to an increase in low cloud reflectivity over the subsiding region, but rather arises from differences in the efficiency of outgoing longwave radiation to space associated with the different relative humidity distributions in the warm and cool regions. The OLR in the hot, moist region is much less efficient than in the cold, dry region, and this difference increases with warming