A Polymer Expansion for the Random Walk on the Permutation Group Associated to the Quantum Heisenberg Ferromagnet

TL;DR

This paper introduces a polymer expansion method for analyzing the heat equation linked to a random walk on the permutation group of a finite lattice, related to the Quantum Heisenberg Ferromagnet, providing a new analytical approach.

Contribution

It presents a novel polymer expansion technique for the heat equation associated with the quantum ferromagnet's random walk on permutation groups.

Findings

Polymer expansion formulated for finite lattice case.

No convergence analysis provided for infinite lattice.

Framework sets foundation for future convergence studies.

Abstract

For a long time one has associated to the Quantum Heisenberg Ferromagnet on a lattice, a random walk on the permutation group of the lattice vertices. We here present a polymer expansion for the solution of the heat equation coupled to the random walk. We work on a finite lattice, there is no question of convergence. We leave to future work bounding terms in the expansion necessary to extend the result to an infinite lattice.