Insertion-tolerance and local isomorphism property for random graphs
Abstract: Any graphed equivalence relation produces a random rooted graph in the sense of Aldous and Lyons, that is, a random variable with values in the space of isomorphism classes of locally finite rooted graphs. We replace this space with the Gromov-Hausdorff space associated with the Cayley graph G of a finitely generated infinite group. This compact (ultra)metric space is endowed with a continuous graphed equivalence relation defined by root moving. Since these two spaces are related by a natural non-expansive map, any random rooted graph with values in G is an example of random rooted graph as defined by Aldous and Lyons. For the abelian free group on two generators, Ghys constructed an unimodular random rooted subtree of G as the continuous hull of a rigid repetitive subtree of G. Other examples of unimodular random rooted graphs obtained from repetitive subgraphs of Cayley graphs are due to Blanc and Lozano Rojo. We show that the continuous hull of any repetitive subgraph in G is negligible for any group-invariant insertion-tolerant bond percolation process on G in the non-trivial supercritical phase. In collaboration with Álvaro Lozano Rojo and Antón C. Vázquez Martínez.
Congreso: Random walks on groups
Trabajo conjunto con Á. Lozano Rojo y A. C. Vázquez Martínez