# On the Probabilistic Representation of the Free Effective Resistance of Infinite Graphs

## TL;DR

This paper characterizes the conditions under which the free effective resistance of infinite graphs can be represented using simple hitting probabilities of the graph's random walk, providing a clear link between resistance and probabilistic behavior.

## Contribution

It offers a complete characterization of when free effective resistance can be expressed via hitting probabilities in infinite graphs, connecting electrical and probabilistic graph properties.

## Key findings

- Identifies conditions for resistance representation in terms of hitting probabilities
- Provides a complete characterization for infinite graphs
- Bridges electrical network theory with probabilistic analysis

## Abstract

We completely characterize when the free effective resistance of an infinite graph can be expressed in terms of simple hitting probabilities of the graphs random walk.