Recursively-constructed couplings have been used in the past for mixing on trees. We show how to extend this technique to non-tree-like graphs such as lattices. Using this method, we obtain the following general result. Suppose that G is a triangle-free graph and that for some Δ ≥ 3, the maximum degree of G is at most Δ. We show that the spin system consisting of q-colourings of G has strong sp...

We consider the anti-ferromagnetic Potts model on the the integer lattice Z. The model has two parameters, q, the number of spins, and λ = exp(−β), where β is “inverse temperature”. It is known that the model has strong spatial mixing if q > 7, or if q = 7 and λ = 0 or λ > 1/8, or if q = 6 and λ = 0 or λ > 1/4. The λ = 0 case corresponds to the model in which configurations are proper q-colouri...

We consider the anti-ferromagnetic Potts model on the the integer lattice Z. The model has two parameters, q, the number of spins, and λ = exp(−β), where β is “inverse temperature”. It is known that the model has strong spatial mixing if q > 7, or if q = 7 and λ = 0 or λ > 1/8, or if q = 6 and λ = 0 or λ > 1/4. The λ = 0 case corresponds to the model in which configurations are proper q-colouri...

We consider the anti-ferromagnetic Potts model on the the integer lattice Z. The model has two parameters, q, the number of spins, and λ = exp(−β), where β is “inverse temperature”. It is known that the model has strong spatial mixing if q > 7, or if q = 7 and λ = 0 or λ > 1/8, or if q = 6 and λ = 0 or λ > 1/4. The λ = 0 case corresponds to the model in which configurations are proper q-colouri...

We consider the anti-ferromagnetic Potts model on the the integer lattice Z. The model has two parameters, q, the number of spins, and λ = exp(−β), where β is “inverse temperature”. It is known that the model has strong spatial mixing if q > 7 or if q = 7 and λ = 0 or λ > 1/8 or if q = 6 and λ = 0 or λ > 1/4. The λ = 0 case corresponds to the model in which configurations are proper q-colouring...

We have observed sub-Poissonian spin correlations generated by collisionally induced spin mixing in a spin-1 Bose-Einstein condensate. We measure a quantum noise reduction of -7 dB (-10 dB corrected for detection noise) below the standard quantum limit for the corresponding coherent spin states. The spin fluctuations are detected as atom number differences in the spin states using fluorescent i...