Georgia Institute of Technology College of Sciences

The asymptotic dimension of metric spaces is an important notion in geometric group theory. The metric spaces considered in this talk are the ones whose underlying spaces are the vertex-sets of (edge-)weighted graphs and whose metrics are the distance function in weighted graphs. A standard compactness argument shows that it suffices to consider the asymptotic dimension of classes of finite weighted graphs. We prove that the asymptotic dimension of any minor-closed family of weighted graphs, any class of weighted graphs of bounded tree-width, and any class of graphs of bounded layered tree-width are at most 2, 1,and 2, respectively. The first result solves a question of Fujiwara and Papasoglu; the second and third results solve a number of questions of Bonamy, Bousquet, Esperet, Groenland, Pirot and Scott. These bounds for asymptotic dimension are optimal and generalize and improve some results in the literature, including results for Riemannian surfaces and Cayley graphs of groups with a forbidden minor.

We introduce a class of random graph processes, which we call flip processes. Each such process is given by a rule which is just a function $\mathcal{R}:\mathcal{H}_k\rightarrow \mathcal{H}_k$ from all labelled $k$-vertex graphs into itself ($k$ is fixed). Now, the process starts with a given $n$-vertex graph $G_0$. In each step, the graph $G_i$ is obtained by sampling $k$ random vertices $v_1,\ldots,v_k$ of $G_{i-1}$ and replacing the induced graph $G_{i-1}[v_1,\ldots,v_k]$ by $\mathcal{R}(G_{i-1}[v_1,\ldots,v_k])$. This class contains several previously studied processes including the Erdos-Renyi random graph process and the random triangle removal.

Given a flip processes with a rule $\mathcal{R}$ we construct time-indexed trajectories $\Phi:\mathcal{W}\times [0,\infty)\rightarrow\mathcal{W}$ in the space of graphons. We prove that with high probability, starting with a large finite graph $G_0$ which is close to a graphon $W_0$, the flip process will follow the trajectory $(\Phi(W_0,t))_{t=0}^\infty$ (with appropriate rescaling of the time).

These graphon trajectories are then studied from the perspective of dynamical systems. We prove that two trajectories cannot form a confluence, give an example of a process with an oscilatory trajectory, and study stability and instability of fixed points.

Joint work with Frederik Garbe, Matas Sileikis and Fiona Skerman.