Felix Pogorzelski - Graphs and spectral approximations, 04th July 2013

The investigation of spectral properties of operators on discrete structures is an essential undertaking of Mathematical Physics. Results in this field give indications concerning conductivity properties of materials according to their molecular structure. Hence, it is a natural question to examine connections between geometry and spectrum on graphs. One issue in this context is the approximation of the normalized spectral distribution function (integrated density of states, IDS) via relative frequencies of matrix spectra. It turns out that, using some kind of ergodic theorems, one can prove results of this kind. The approximation can be obtained along graph sequences converging in a locally statistical sense. In general, the limit object will not be a single graph, but a probability distribution of countable graphs.To obtain stronger convergence, more assumptions on the sequences are necessary. Therefore, we draw our attention to the the concept of hyperfiniteness for graph sequences. For those objects, a general almost-additive convergence theorem can be proven. We explain the connection to uniform convergence of the IDS.