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We give a near-linear time sampler for the Gibbs distribution of theferromagnetic Ising models with edge activities $\boldsymbol{\beta} > 1$ andexternal fields $\boldsymbol{\lambda}<1$ (or symmetrically,$\boldsymbol{\lambda}>1$) on general graphs with bounded or unbounded maximumdegree.  Our algorithm is based on the field dynamics given in [CFYZ21]. We prove thecorrectness and efficiency of our algorithm by establishing spectralindependence of distribution of the random cluster model and the rapid mixingof Glauber dynamics on the random cluster model in a low-temperature regime,which may be of independent interest.

A Near-Linear Time Sampler for the Ising Model with External Field