Linear time mixing and pre-cutoff for oriented kinetically constrained models
Abstract
We establish linear time precutoff for oriented kinetically constrained models (KCMs) on a d-dimensional box of side length n, whenever the finite-volume spectral gap is bounded away from zero uniformly in n. A typical example is the North-East model, a $0−1$ spin system on the two-dimensional integer lattice that evolves according to the following rule: whenever a site's southerly and westerly nearest neighbours have spin $0$, with rate one it resets its own spin by tossing a (1−q)-coin; at all other times its spin remains frozen. This settles, in the oriented case, a conjecture which states that there is linear time precutoff for all KCMs in the (open) ergodic parameter regime (q>qc). The result has been shown recently, for general update families, in a perturbative regime (q close to $1$). The previous general upper bound on the mixing time for oriented update families, in the full ergodic regime, was O(n log n). The proof combines a spectral estimate for a certain killed process with a graphical coupling to a stationary process and a backward witness path argument, exploiting the orientation of the constraints, to control the probability of failing to couple.
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BibTeX
@misc{chleboun2026linear,
title = {{Linear time mixing and pre-cutoff for oriented kinetically constrained models}},
author = {Paul Chleboun},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30585},
eprint = {2609.30585},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30585},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Linear time mixing and pre-cutoff for oriented kinetically constrained models
AU - Paul Chleboun
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30585
UR - https://arxiv.org/abs/2609.30585
AB - We establish linear time precutoff for oriented kinetically constrained models (KCMs) on a d-dimensional box of side length n, whenever the finite-volume spectral gap is bounded away from zero uniformly in n. A typical example is the North-East model, a $0−1$ spin system on the two-dimensional integer lattice that evolves according to the following rule: whenever a site's southerly and westerly nearest neighbours have spin $0$, with rate one it resets its own spin by tossing a (1−q)-coin; at all other times its spin remains frozen. This settles, in the oriented case, a conjecture which states that there is linear time precutoff for all KCMs in the (open) ergodic parameter regime (q>q_c). The result has been shown recently, for general update families, in a perturbative regime (q close to $1$). The previous general upper bound on the mixing time for oriented update families, in the full ergodic regime, was O(n log n). The proof combines a spectral estimate for a certain killed process with a graphical coupling to a stationary process and a backward witness path argument, exploiting the orientation of the constraints, to control the probability of failing to couple.
ER -