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Combinatorial identities from an inhomogeneous Ising chain

Jessica Jay, Benjamin Lees

Abstract

We study a family of inhomogeneous Ising chain models along with an equivalent family of nearest neighbour particle systems. By the correspondence between the two families we prove identities of combinatorial significance relating to certain integer and Frobenius partitions. In particular, for certain parameter values we see that one of our identities relates to generating functions for overpartitions. Using the identities we give a surprising product form of the partition function for an Ising chain with homogeneous interaction and an inhomogeneous external field. We also use the connection between the Ising chain and particle system to find interesting long-range reversible dynamics for the particle system that do not have a product form stationary measure.

Combinatorial identities from an inhomogeneous Ising chain

Abstract

We study a family of inhomogeneous Ising chain models along with an equivalent family of nearest neighbour particle systems. By the correspondence between the two families we prove identities of combinatorial significance relating to certain integer and Frobenius partitions. In particular, for certain parameter values we see that one of our identities relates to generating functions for overpartitions. Using the identities we give a surprising product form of the partition function for an Ising chain with homogeneous interaction and an inhomogeneous external field. We also use the connection between the Ising chain and particle system to find interesting long-range reversible dynamics for the particle system that do not have a product form stationary measure.
Paper Structure (15 sections, 21 theorems, 137 equations, 13 figures, 3 tables)

This paper contains 15 sections, 21 theorems, 137 equations, 13 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Results
  3. An Asymmetric Blocking Ising process under Kawasaki dynamics
  4. The conserved quantity for Kawasaki dynamics on blocking configurations
  5. Expression for $\mu^c_J(\{N=n\})$ using the shift operator
  6. Transferring dynamics to a family nearest neighbour particle systems
  7. Identities found by equating measures for the corresponding families
  8. An expression for the partition functions in terms of runs of consecutive spins
  9. Combinatorial interpretation of identities
  10. Combinatorial Identity for $J(i)\equiv1$
  11. Combinatorial Identity for $J(i)=i$
  12. The Ising model with long range interactions and Kawasaki Dynamics
  13. Transferring dynamics to a particle system
  14. Restricted long-range dynamics
  15. Natural reversible dynamics for the particle system

Key Result

Theorem 1.1

For $Q\in(0,1)$, $z>0$, and $y \in (0,1]$, Where, with $\ell_0=-1$ and $m_0=0$.

Figures (13)

  • Figure 3.1: An example of the bijection $T^{-1}$.
  • Figure 5.1: The integer partition $(8,7,7,3,1,1)$ and its Frobenius partition.
  • Figure 5.2: Examples of the number of distinct sizes in certain integer partitions and the equivalent value for the corresponding Frobenius partitions.
  • Figure 5.3: Examples of the Wright bijections $\phi_3$ and $\phi_{-3}$ (adapted from MDJ Figure 5).
  • Figure 5.4: Examples of the Wright bijections $\phi_3$ and $\phi_{-3}$ (adapted from MDJ Figure 5).
  • ...and 8 more figures

Theorems & Definitions (46)

  • Theorem 1.1
  • Corollary 1.1
  • Theorem 1.2
  • Remark 2.1
  • Lemma 2.2
  • Remark 2.3
  • proof
  • Remark 2.4
  • Proposition 2.5
  • Lemma 2.6
  • ...and 36 more