# Kramers–Wannier Duality and Random-Bond Ising Model

## TL;DR

This paper introduces a new method for calculating the Ising model's free energy using planar graphs and their duals, revealing insights into Kramers–Wannier duality and the Random-Bond Ising Model.

## Contribution

A novel combinatorial approach that expresses the Ising model's free energy via determinants of operators on planar graphs and their duals.

## Key findings

- The exact free energy is derived using determinants of ordered and disordered operators.
- The approach explicitly demonstrates Kramers–Wannier duality on planar graphs.
- Implications for the Random-Bond Ising Model are clarified through the derived formula.

## Abstract

We present a new combinatorial approach to the Ising model incorporating arbitrary bond weights on planar graphs. In contrast to existing methodologies, the exact free energy is expressed as the determinant of a set of ordered and disordered operators defined on a planar graph and the corresponding dual graph, respectively, thereby explicitly demonstrating the Kramers–Wannier duality. The implications of our derived formula for the Random-Bond Ising Model are further elucidated.