## David Sivakoff : Polluted Bootstrap Percolation in Three Dimensions

- Probability ( 231 Views )In r-neighbor bootstrap percolation, the vertices of Z^d are initially occupied independently with probability p and empty otherwise. Occupied vertices remain occupied forever, and empty vertices iteratively become occupied when they have at least r occupied neighbors. It is a classic result of van Enter (r=d=2) and Schonmann (d>2 and r between 2 and d) that every vertex in Z^d eventually becomes occupied for any initial density p>0. In the polluted bootstrap percolation model, vertices of Z^d are initially closed with probability q, occupied with probability p and empty otherwise. The r-neighbor bootstrap rule is the same, but now closed vertices act as obstacles, and remain closed forever. This model was introduced 20 years ago by Gravner and McDonald, who studied the case d=r=2 and proved a phase transition exists for this model as p and q tend to 0. We prove a similar phase transition occurs when d=r=3, and we identify the polynomial scaling between p and q at which this transition occurs for the modified bootstrap percolation model. For one direction, our proof relies on duality methods in Lipschitz percolation to find a blocking structure that prevents occupation of the origin. The other direction follows from a rescaling argument, and the recent results of Holroyd and Gravner for d>r=2. This is joint work with Holroyd and Gravner.

## John McSweeney : A Nonuniform Stochastic Coalescent Process with applications to Biology and Computer Science

- Probability ( 225 Views )Viewed forwards in time, a population reproducing according to some random mechanism can be thought of as a branching process. What if it is viewed backwards? We can take a sample of individuals from the current generation and trace their genealogy backwards, and for instance find their most recent common ancestor; this is known as a coalescent process. If we know a population's random mating process, but have no actual data as to what the phylogenetic tree looks like, how do we derive the distribution of the time until its most recent common ancestor? I will discuss a variant on the classical Wright-Fisher reproductive model and deduce some parameter thresholds for emergence of different qualitative features of the tree. An isomorphic problem may also be useful in computer science for bounding the running time of certain random sampling algorithms.

## Robin PEMANTLE : Zeros of random analytic functions and their derivatives

- Probability ( 207 Views )I will discuss a series of results concerning the effect of the derivative operator on the locations of the zeros of a random analytic function. Two models are considered. In the first, the zeros are chosen IID from some measure on the complex plane. In the second, the zeros are chosen to be a Poisson point process on the real line. Repeated differentiation results in a nearly deterministic zero set.

## Laurie Field : Relating variants of SLE using the Brownian loop measure

- Probability ( 205 Views )In this talk I will discuss a framework for transforming one variant of the SchrammLoewner evolution (SLE) into another. The main tool in this approach is the Brownian loop measure. A simple case is to relate the reversal of radial SLE to whole-plane SLE, which looks the same locally. Writing the formula one might naïvely expect fails, because the loop measure term is infinite. In joint work with Greg Lawler, we show that there is a finite normalized version of the loop measure term, and that with this change, the naïve formula relating the two SLEs becomes correct.