Dongming Hua | Searcharxiv

arXiv2025

Volume Entropy Rigidity for Random Groups at Low Densities

We study the rigidity of the volume entropy for weighted word metrics on hyperbolic groups, building on a recent convexity result due to Cantrell-Tanaka. Using ideas from small cancellation theory, we give conditions under which a hyperbolic group admits a unique normalized weight minimizing the entropy. Moreover, we show that these conditions are generic for random groups at small densities, and that the unique minimizer of such a generic group is arbitrarily close to the uniform weight.

math.GR↗

arXiv2023

Local behavior of the Eden model on graphs and tessellations of manifolds

The Eden Model in $\mathbb{R}^n$ constructs a blob as follows: initially a single unit hypercube is infected, and each second a hypercube adjacent to the infected ones is selected randomly and infected. Manin, Roldán, and Schweinhart investigated the topology of the Eden model in $\mathbb{R}^{n}$ by considering the possible shapes which can appear on the boundary. In particular, they give probabilistic lower bounds on the Betti numbers of the Eden model. In this paper, we prove analogous results for the Eden model on any infinite, vertex-transitive, locally finite graph: with high probability as time goes to infinity, every "possible" subgraph (with mild conditions on what "possible" means) occurs on the boundary of the Eden model at least a number of times proportional to an isoperimetric profile of the graph. Using this, we can extend the results about the topology of the Eden model to non-Euclidean spaces, such as hyperbolic $n$-space and universal covers of certain Riemannian manifolds.

math.PR↗