October 5, 2012
Location: Janskerkhof 15a (Utrecht), room 106
Non-backtracking lace expansion for nearest-neighbor models
In this talk we introduce the non-backtracking lace expansion (NoBLE), that we have developed to enhance known results in the nearest-neighbor setting on the Zd-lattice. We will first review the concept of mean-field behavior for self-avoiding walks and percolation. Then we informally describe the classical lace expansion and NoBLE that can be used to prove mean-field behavior in high dimensions. We work out the differences between the two techniques and show heuristically how they are able to prove mean-field behavior. In the last part we will see a Mathematica program that implements the lace expansion analysis for self-avoiding walk and percolation.
Ancestry in locally regulated populations and a random walk in a dynamic random environment
The spatial embeddings of genealogies in models with fluctuating population sizes and local regulation are relatively complicated random walks in a space-time dependent random environment. We discuss regeneration approaches to study their long-time behaviour and implications for spatial type distributions. A simple guiding example is the discrete time contact process, with an ancestral lineage given by a directed walk on an oriented percolation cluster.
Based on joint work with Jiří Černý, Andrej Depperschmidt and Nina Gantert.