Trapping in Elliptic Environments and Renewal Structures for Tree-Building Walks

We discuss two distinct classes of random walks in random or dynamic environments. First, we consider random walks in i.i.d. elliptic environments that are not uniformly elliptic. In dimension $d=2$, we introduce a geometric framework to classify how the walk becomes trapped in edges, wedges, or squares. This leads to explicit, computable criteria on the exit times from a box which ensure that local trapping does not inhibit ballistic behavior. This part represents joint work with Alejandro Ramírez. Second, we transition to the Tree Builder Random Walk (TBRW), a model where the walker attaches a random number of leaves to its current position at each step. By introducing a renewal framework analogous to the Sznitman–Zerner structure for RWRE, we establish fundamental limit theorems under an i.i.d. hypothesis, including the Strong Law of Large Numbers, an Invariance Principle, and the continuity of the asymptotic speed with respect to the environment distribution. The talk will be based on the papers: R.R. Renewal structure of the tree builder random walk. SPA -2025 Alejandro F. Ramírez, R. R. Computable criteria for ballisticity of random walks in elliptic random environment. Electronic Journal of Probability - 2022