### Abstract

We note that a tridiagonal matrix representation of the algebra of the partially asymmetric exclusion process (PASEP) lends itself to interpretation as the transfer matrix for weighted Motzkin lattice paths. A continued-fraction ('J fraction') representation of the lattice-path-generating function is particularly well suited to discussing the PASEP, for which the paths have height-dependent weights. We show that this not only allows a succinct derivation of the normalization and correlation lengths of the PASEP, but also reveals how finite-dimensional representations of the PASEP algebra, valid only along special lines in the phase diagram, relate to the general solution that requires an infinite-dimensional representation.

Original language | English |
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Article number | 325002 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 42 |

Issue number | 32 |

DOIs | |

Publication status | Published - 2009 |

### Bibliographical note

The full text is available free from the link given. The published version can be found at http://dx.doi.org/10.1088/1751-8113/42/32/325002.### Keywords

- phase transitions: general studies
- matrix theory
- nonequilibrium and irreversible thermodynamics
- combinatorics
- graph theory
- lattice theory and statistics (Ising
- Potts
- etc.)

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## Cite this

Blythe, R. A., Janke, W., Johnston, D. A., & Kenna, R. (2009). Continued fractions and the partially asymmetric exclusion process.

*Journal of Physics A: Mathematical and Theoretical*,*42*(32), [325002]. https://doi.org/10.1088/1751-8113/42/32/325002