Mixing times of spin systems on dynamical percolation
2026-07-02 • Discrete Mathematics
Discrete Mathematics
AI summaryⓘ
The authors study how fast a system of spins (tiny magnetic moments) changes over time when the connections between them randomly open and close on a grid shaped like a torus. Each connection flips open or closed at some rate, and each spin updates based on its neighbors connected by open edges. They find that when the probability of an edge being open is below a critical threshold, and the edge update rate is slow enough, the system mixes in a time proportional to the logarithm of the grid size divided by the edge update rate. Their proof uses a special method to link small parts of the system when the environment is behaving well, even though the whole process doesn’t follow simple reversible rules.
mixing timestochastic spin systemsGlauber dynamicsdynamical percolationd-dimensional toruscritical probability p_cMarkov chainnon-reversible processcoupling methodnearest-neighbour interactions
Authors
Alexandre Stauffer, Oskar Vavtar
Abstract
We study the mixing times of stochastic spin systems corresponding to nearest-neighbour Glauber dynamics on dynamical percolation, defined on $d$-dimensional torus of side-length $N$. In this model, the status of each edge (open or closed) updates independently at rate $λ>0$, according to $\mathrm{Ber}(p)$ samples. Simultaneously, the spin of each site updates at rate $1$ according to Glauber dynamics on the environment restricted to open edges. We show that for a relatively general class of nearest-neighbour systems, as long as $p<p_c(d)$, for any temperature, if $λ$ is sufficiently small, the mixing time is of order $\frac{\log N}λ$. This Markov chain is non-reversible, and the proof is obtained by developing a particular coupling that couples together local configurations whenever the environment behaves well.