Lesekurs: Local weak limits of graphs

If you wish to participate contact me by email no later than 22 April 2026

Local weak limits (or Benjamini-Schramm limits) are a framework in graph theory for studying the limiting behaviour of large graphs by focusing on the local neighbourhoods around randomly chosen vertices. Instead of requiring global convergence (which is often too strong or impractical), they capture how a “typical” finite-radius neighbourhood looks as the graph size grows. This is especially important for sparse graphs, where classical dense graph limit theories (like graphons) do not apply well. Local weak limits connect finite graphs to infinite random rooted graphs, enabling the use of probabilistic and analytical tools. They are widely used in areas such as random graph theory, statistical physics, and network science to understand properties like percolation, phase transitions, and algorithmic behaviour.

We will read parts of Random Graphs and Complex Networks (vol. 2) by Remco van der Hofstad. The main goal is to understand the theoretical concepts of Section~2 and their applications to Erdös-Rényi random graphs. If time permits we will also study the consequences for suitable spectral measures of graph Laplacians or adjacency operators as in Spectrum of random graphs by Charles Bordenave (Panoramas et Synthèses, vol. 53 (2017), Advanced topics in random matrices, pp. 91 – 150).

MSc students of Mathematics

Measure and probability theory, basic notions of graph theory; for the last part also basics of spectral theory