Rapid mixing of Glauber dynamics via spectral independence for all degrees

Optimal mixing for two-state anti-ferromagnetic spin systems

“…By choosing a suitable constant gap , the MLSI for the original near-critical distribution is reduced, by losing a constant factor, to the MLSI for the magnetized distribution * that falls into a subcritical regime, where the minimum MLS constant GD min ( * ) is easier to analyze. A similar boosting theorem for the Poincaré constant (spectral gap) was established in [CFYZ21b], essentially by assuming the spectral independence part of Condition 1.10. Here we prove a similar boosting for the MLSI by further assuming the marginal stability.…”

“…-unique for all 1 ≤ ≤ Δ − 1) assumed in e.g. [LLY13,CLV20,CLV21,CFYZ21b] for spin systems with Δ-bounded maximum degree .…”

“…The notion of spectral independence was introduced by Anari, Liu and Oveis Gharan in [ALO20]. We use the absolute version of the spectral independence considered in [FGYZ21,CFYZ21b].…”

“…An ( = 0) variant of Condition 1.10 (1) was used in [CFYZ21b] and called complete spectral independence.…”

“…Remark 1.12 (applications to spin systems). When applying Theorem 1.11 to anti-ferromagnetic two-spin systems, as in [CFYZ21b], we can first preprocess the distribution by properly flipping the roles of spins in {+1, −1} for each vertex, so that after the preprocessing, the distribution will only get "less critical" by decreasing the local field at every vertex. We then formally verify the spectral independence and the marginal stability properties in Condition 1.10 for the flipped distribution assuming the uniqueness.…”