A remark on monotonicity for the Glauber dynamics on finite graphs
Properties of the spectral gap in the trasition matrix of the Glauber dynamics are investigated, with a focus on a monotonicity hypothesis. Its general validity is still an open problem. It is known that the gap may be monotone decreasing in each coupling constant of the Hamiltonian, for cycles of any length. The present author departs from correlation inequalities (GKS and GHZ) for spin systems to verify whether the single spin expectation for a fixed time in the relaxation process starting from all-up configuration is monotone increasing in each coupling constant. If that would hold true the monotonicity of the spectral gap would follow. The answer is negative (counter example).
scientific article; zbMATH DE number 5690872