Rapid mixing from spectral independence beyond the Boolean domain
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We extend the notion of spectral independence (introduced by Anari, Liu, and Oveis Gharan [ALO20]) from the Boolean domain to general discrete domains. This property characterises distributions with limited correlations, and implies that the corresponding Glauber dynamics is rapidly mixing. As a concrete application, we show that Glauber dynamics for sampling proper q-colourings mixes in polynomial-time for the family of triangle-free graphs with maximum degree Δ provided q≥ (α^*+δ)Δ where α^*≈ 1.763 is the unique solution to α^*=(1/α^*) and δ>0 is any constant. This is the first efficient algorithm for sampling proper q-colourings in this regime with possibly unbounded Δ. Our main tool of establishing spectral independence is the recursive coupling by Goldberg, Martin, and Paterson [GMP05].
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