Diffusion on an Ising chain with kinks

TL;DR

This paper counts the number of possible transition histories between two energy states in an Ising chain with kinks, providing explicit formulas and asymptotic analysis relevant for quantum Hamiltonian dynamics.

Contribution

It introduces a combinatorial approach to counting histories in the Ising model with kinks, linking it to permutation avoidance and quantum dynamics.

Findings

Derived explicit generating functions for the counts

Computed asymptotic behavior of the number of histories

Connected combinatorial results to quantum Hamiltonian dynamics

Abstract

We count the number of histories between the two degenerate minimum energy configurations of the Ising model on a chain, as a function of the length n and the number d of kinks that appear above the critical temperature. This is equivalent to count permutations of length n avoiding certain subsequences depending on d. We give explicit generating functions and compute the asymptotics. The setting considered has a role when describing dynamics induced by quantum Hamiltonians with deconfined quasi-particles.