Near-optimal O~(n) mixing for Glauber dynamics below the λ_c threshold

Establish a near-optimal \~O(n) upper bound on the mixing time of Glauber dynamics for (β,γ,λ)-ferromagnetic two-spin systems on general graphs in the regime β ≤ 1 < γ, βγ > 1, and λ < λ_c(β,γ) := (γ/β)^{sqrt(βγ)/(sqrt(βγ) − 1)}, starting from an arbitrary initial configuration.

Background

The paper proves near-linear (up to polylogarithmic factors) mixing for Glauber dynamics when λ < λ_0 := sqrt(γ/β), and polynomial (n^2–n^3 up to polylog) mixing when extended to the larger regime λ < λ_c(β,γ). The authors highlight technical obstacles that prevent directly extending their λ < λ_0 analysis to the regime λ < λ_c.

Achieving an ~O(n) mixing time in the full algorithmic regime λ < λ_c would be optimal up to logarithmic factors and would close the remaining gap left by the current techniques, which rely on a “typical-case” ASSM framework and spectral-independence-based boosting.