Information Percolation and cutoff for Potts Glauber Dynamics
Abstract
We prove cutoff with an O(1) window for the continuous-time heat-bath Glauber dynamics of the ferromagnetic q-state Potts model on the discrete torus Λn=(ℤ/nℤ)d at sufficiently high temperature. More precisely, for every fixed d≥2 and q≥3, there exists β0=β0(d,q)>0 such that, for 0<β<β0, the cutoff occurs in an optimal O(1) window centered at t⋆=t⋆(n):=1/2rlog |Λn|, where r∈(0,1) is the exact exponential decay rate of the one-site color bias. Thus, the mixing time is determined up to an additive O(1). It is characterized by the time at which the macroscopic color-density bias from the monochromatic initial condition enters the scale of equilibrium fluctuations. Moreover, our proof shows that the monochromatic initial condition uniquely maximizes the color bias. This is the first implementation of information percolation to prove cutoff for a non-monotone spin system. On a transitive graph, information percolation naturally gives an upper bound on the mixing time of one half of the grand coupling time which is sharp for the Ising model. However, for many non-monotone systems including the Potts model, this misses additional averaging over the update histories giving an upper bound that is off by a constant multiplicate factor. We develop a new framework that treats the information percolation histories as random polymers and show how to average over them to give the effective bias from the initial condition. This accounts for all the additional mixing, recovering a sharp upper bound and cutoff. We believe this approach will be applicable in a much wider range of settings.
Related
- Rapid mixing of Potts Glauber dynamics throughout uniqueness
- Cutoff Phenomenon for the Glauber-Misanthrope process in a perturbative regime
- Analyticity of the correlation length in subcritical FK-percolation and Potts models
- Glauber Dynamics for Sampling Colorings: Rapid Mixing above (1+1/e)Δ via Hidden Hard-Core Models
- Mixing profile for Glauber dynamics of the discrete Gaussian Free Field starting from super-harmonic functions
BibTeX
@misc{yang2026information,
title = {{Information Percolation and cutoff for Potts Glauber Dynamics}},
author = {Seoyeon Yang and Allan Sly},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.26259},
eprint = {2608.26259},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.26259},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Information Percolation and cutoff for Potts Glauber Dynamics
AU - Seoyeon Yang
AU - Allan Sly
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2608.26259
UR - https://arxiv.org/abs/2608.26259
AB - We prove cutoff with an O(1) window for the continuous-time heat-bath Glauber dynamics of the ferromagnetic q-state Potts model on the discrete torus Λ_n=(ℤ/nℤ)^d at sufficiently high temperature. More precisely, for every fixed d≥2 and q≥3, there exists β_0=β_0(d,q)>0 such that, for 0<β<β_0, the cutoff occurs in an optimal O(1) window centered at t_⋆=t_⋆^(n):=1/2rlog |Λ_n|, where r∈(0,1) is the exact exponential decay rate of the one-site color bias. Thus, the mixing time is determined up to an additive O(1). It is characterized by the time at which the macroscopic color-density bias from the monochromatic initial condition enters the scale of equilibrium fluctuations. Moreover, our proof shows that the monochromatic initial condition uniquely maximizes the color bias. This is the first implementation of information percolation to prove cutoff for a non-monotone spin system. On a transitive graph, information percolation naturally gives an upper bound on the mixing time of one half of the grand coupling time which is sharp for the Ising model. However, for many non-monotone systems including the Potts model, this misses additional averaging over the update histories giving an upper bound that is off by a constant multiplicate factor. We develop a new framework that treats the information percolation histories as random polymers and show how to average over them to give the effective bias from the initial condition. This accounts for all the additional mixing, recovering a sharp upper bound and cutoff. We believe this approach will be applicable in a much wider range of settings.
ER -