2026-09-14
On a weakly coupled Ising star, lazy Metropolis mixes in at least d·exp{cβ(d−1)}; averaging over global spin reversal cuts that to O(d²(1+β)).
Single-site Metropolis (Glauber) dynamics for the Ising model often stall at low temperature. Global spin reversal splits the state space into two high-mass regions; getting from all-plus to all-minus means crossing an exponentially thin interface, so the mixing time tracks that exponential.
This 11-page note pins the phenomenon to a toy that can be computed by hand: an Ising model on a star, plus extra signed pairwise couplings among the leaves. The point is an explicit example where group averaging turns a torpid chain into a rapidly mixing one.
The PDF lives only on ResearchGate and could not be fetched. What follows is from the title-page abstract; proof details are in the unretrieved full text.
The base kernel \(P\beta\) is lazy single-site Metropolis at inverse temperature \(\beta\ge 0\) on \(\{-1,+1\}^d\). Averaging uses the two-element group of global spin reversal, in two concrete kernels:
Local Metropolis rarely crosses the energy barrier; a global reversal maps one mode onto the other in one shot. Upper bounds in the abstract come from Wu's Dobrushin inequality plus a projection/restriction decomposition of the chain.
If the absolute row sums of the leaf-interaction matrix are at most \(\kappa\le 1/2\), then for \(\beta\ge 1\) the worst-case total-variation mixing time of \(P\beta\) is at least on the order of \(d\exp\{c\beta(d-1)\}\) for a universal \(c>0\). The exponential sits on the number of leaves.
Under the same condition both \(GP\beta G\) and \((P\beta+G)/2\) mix in \(O(d^2(1+\beta))\). Temperature enters only linearly, dimension only quadratically. The exponential bottleneck is gone.
The abstract reports no simulations and no comparison with parallel tempering or Swendsen–Wang. This is a theorem, not a table.
For people who write MCMC, this is a worked drawing of what group averaging actually does. Longer papers in the same line recast HMC and PDMP as averaged kernels; this note shrinks the story to a star graph you can compute. If the target is nearly invariant under an involution (global flip, sign flip, mode swap), wrapping a local kernel with an orbit Gibbs step may beat retuning the local kernel.
It is not a general accelerator. \(\kappa\le 1/2\) forces weak leaf coupling, and a star is not a lattice or a fully connected graph.
The body of the paper was not retrieved. How the Dobrushin bound is checked, how laziness is set, and how the projection chain is controlled are not in the abstract. Whether the exponential lower bound returns once \(\kappa>1/2\) is open in the text we have. The model is far from lattice Ising or Boltzmann machines. Computational cost of the sandwich versus the additive mix is not compared.