Dimer model and random lattice permutations with general weights
Abstract
The dimer model and random lattice permutations are two fundamental objects at the interface of probability, combinatorics, and mathematical physics. We study these models on finite periodic boxes in ℤd within a common framework. For the dimer model, edges connecting arbitrary vertices carry a weight which depends on their relative displacement and dimer configurations are weighted through their occupied edges. Superimposing two independent perfect matchings gives the double-dimer model, whose configurations are collections of disjoint loops. Permutations, instead, are weighted through the spatial displacement of their jumps. For broad classes of weights of finite or infinite range we prove long-range order and the occurrence of macroscopic loops. This extends nearest-neighbour results to arbitrary-range edges and jumps. In particular, long-range weights yield long-range order and macroscopic loops already in dimensions d=1,2. In dimension two, this behaviour is qualitatively different from that of the nearest-neighbour model. We complement these results with sharp absence criteria.
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BibTeX
@misc{klippel2026dimer,
title = {{Dimer model and random lattice permutations with general weights}},
author = {Andreas Klippel and Lorenzo Taggi and Wei Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29671},
eprint = {2609.29671},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29671},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Dimer model and random lattice permutations with general weights
AU - Andreas Klippel
AU - Lorenzo Taggi
AU - Wei Wu
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.29671
UR - https://arxiv.org/abs/2609.29671
AB - The dimer model and random lattice permutations are two fundamental objects at the interface of probability, combinatorics, and mathematical physics. We study these models on finite periodic boxes in ℤ^d within a common framework. For the dimer model, edges connecting arbitrary vertices carry a weight which depends on their relative displacement and dimer configurations are weighted through their occupied edges. Superimposing two independent perfect matchings gives the double-dimer model, whose configurations are collections of disjoint loops. Permutations, instead, are weighted through the spatial displacement of their jumps. For broad classes of weights of finite or infinite range we prove long-range order and the occurrence of macroscopic loops. This extends nearest-neighbour results to arbitrary-range edges and jumps. In particular, long-range weights yield long-range order and macroscopic loops already in dimensions d=1,2. In dimension two, this behaviour is qualitatively different from that of the nearest-neighbour model. We complement these results with sharp absence criteria.
ER -