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**A remark on monotonicity for the Glauber dynamics on finite graphs.**
*(English)*
Zbl 1192.82050

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).

Reviewer: Piotr Garbaczewski (Opole)

### MSC:

82C20 | Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics |

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

### Keywords:

Ising model; Glauber dynamics; spectral gap; monotonicity conjecture; correlation inequalities; spin expectation
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\textit{T. Shirai}, Proc. Japan Acad., Ser. A 86, No. 2, 33--37 (2010; Zbl 1192.82050)

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### References:

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