Let $\Gamma$ be a finitely generated group, and denote by $G$ the Cayley graph of $\Gamma$. Denote by $d_R$ the resistance distance metric on this graph. The resistance distance metric between the vertices $x$ and $y$ is the effective resistance between $x$ and $y$ in the electrical network obtained from $G$ by replacing every edge with $1 \Omega$ resistor.

There are many tools to calculate the resistance distance between vertices in a a general graph. It is also known that if we consider an infinite graph of bounded degree then the resistance distance between two vertices is a limit of the effectice resistances in larger and larger finite subgraphs exhausting the original graph.

I am interesited in the question: how good can I approximate the resistance distance between two vertices in an infite Cayley graph by looking at a ball of a finite radius in this graph? What is the speed of convergence?

For me the most interesting is the case of a Cayley graph of a hyperbolic group. The examples show that this convergence is very fast: for a free group we obtain the exact value of the resistance just if we take a ball in the Cayley graph cointaing our two vertices.

Do You know any theorems or papers that could help?