Speaker: Simon Schwarz (Goettingen)
Title: Heat kernel asymptotics for scaling limits of isoradial graphs

Abstract: We consider the asymptotics of the discrete heat kernel on isoradial graphs for the case where the time and the edge lengths tend to zero simultaneously. Depending on the asymptotic ratio between time and edge lengths, we show that two different regimes arise, corresponding to the short-time asymptotics of the heat kernel on (i) Euclidean spaces and (ii) on graphs.
 
Speaker: Anja Sturm (Goettingen)  
Title: On cooperative branching with death on trees

Abstract: We present a  cooperative branching model on the binary tree. This is a random cellular automaton  in which each node of the tree can have the value zero or one. In each time step, each node is updated independently, with probability p the node becomes zero, with probability 1-p the node takes the value one if either its previous value was one or the two nodes above it both had the value one. The system is started in nice initial laws but in particular in product law with intensity q. Here, the special case p=0 is directed bootstrap percolation on the binary tree, which has been studied before.
We are interested in the long time behavior and invariant laws. We show that there is a nontrivial phase transition in p such that above the critical value the invariant laws are trivial, and study the invariant laws below the critical value. We also show that the stationary model can be viewed as the limit of finite volume models and as a recursive tree process (RTP) on time sequences, and make connections to earlier work on RTP and cooperative branching models.
This is work in progress with Jan Swart and Natalia Cardona-Tobon.