Restrictions of some reinforced processes to subgraphs

TL;DR

This paper demonstrates that restricting certain reinforced stochastic processes to subgraphs results in mixtures of similar processes, with applications to Markov chains and reinforced random walks on subdivided graphs.

Contribution

It establishes that vertex-reinforced jump processes and hyperbolic sigma models retain their mixture structure under restriction to subgraphs, extending understanding of their behavior.

Findings

Restrictions produce mixtures of reinforced processes.

Results apply to subdivided graphs with bounded degree.

Discrete-time processes are mixtures of positive recurrent Markov chains.

Abstract

We prove that the restriction of the vertex-reinforced jump process to a subset of the vertex set is a mixture of vertex-reinforced jump processes. A similar statement holds for the non-linear hyperbolic supersymmetric sigma model. This is then applied to vertex-reinforced jump processes on subdivided versions of graphs of bounded degree, where every edge is replaced by a finite sequence of edges. We prove that discrete-time processes associated to suitable corresponding restrictions are mixtures of positive recurrent Markov chains. We also deduce a similar statement for edge-reinforced random walks.