Abstract

Consider the vertex-reinforced jump process on the complete graph with vertex set Zd, d=1,2,...,  and edge weights Wij=w(|i-j|), i,j in Zdwith a strictly positive decreasing weight function w. If w satisfies a summability condition and a suitable lower bound, then the vertex-reinforced jump process is a.s. transient. Using the representation of the discrete time process associated with the vertex-reinforced jump process on finite boxes as a random walk in random conductances, the key estimate consists in a bound for (cosh ui)m for the corresponding H2|2 model. We compare this H2|2 model withanother H2|2 model with hierarchical interactions. The task of studying cosh ui in the last model can be reduced to studying it in an effective H2|2 model on a line graph with inhomogeneous interactions. The proof of some estimates is inspired by work of Disertori-Spencer-Zirnbauer 2010.  The talk is based on https://arxiv.org/abs/2305.07359, which is joint work with Margherita Disertori and Franz Merkl.