We prove an optimal mixing time bound on the single-site update Markov chain known as the Glauber dynamics or Gibbs sampling in a variety of settings. Our work presents an improved version of the spectral independence approach of Anari et al. (2020) and shows O(nn) mixing time on any n-vertex graph of bounded degree when the maximum eigenvalue of an associated influence matrix is bounded. As an application of our results, for the hard-core model on independent sets weighted by a fugacity \lambda, we establish O(nn) mixing time for the Glauber dynamics on any n-vertex graph of constant maximum degree \Delta when \lambda<\lambda_c(\Delta) where \lambda_c(\Delta) is the critical point for the uniqueness/non-uniqueness phase transition on the \Delta-regular tree. More generally, for any antiferromagnetic 2-spin system we prove O(n\log{n) mixing time of the Glauber dynamics on any bounded degree graph in the corresponding tree uniqueness region. Our results apply more broadly; for example, we also obtain O(nn) mixing for q-colorings of triangle-free graphs of maximum degree \Delta when the number of colors satisfies q \geq \alpha \Delta + 1 } ) mixing time of the Glauber dynamics on any bounded degree graph in the corresponding tree uniqueness region. Our results apply more broadly; for example, we also obtain O(nn) mixing for q-colorings of triangle-free graphs of maximum degree \Delta when the number of colors satisfies q > \alpha \Delta } where \alpha \approx 1.763, and O(mn) mixing for generating random matchings of any graph with bounded degree and m edges.
We prove an optimal mixing time bound on the single-site update Markov chain known as the Glauber dynamics or Gibbs sampling in a variety of settings. Our work presents an improved version of the spectral independence approach of Anari et al. (2020) and shows O(nn) mixing time on any n-vertex graph of bounded degree when the maximum eigenvalue of an associated influence matrix is bounded. As an application of our results, for the hard-core model on independent sets weighted by a fugacity \lambda, we establish O(nn) mixing time for the Glauber dynamics on any n-vertex graph of constant maximum degree \Delta when ) mixing time of the Glauber dynamics on any bounded degree graph in the corresponding tree uniqueness region. Our results apply more broadly; for example, we also obtain O(nn) mixing for q-colorings of triangle-free graphs of maximum degree \Delta when the number of colors satisfies q \geq \alpha \Delta + 1 } \lambda<\lambda_c(\Delta) where \lambda_c(\Delta) is the critical point for the uniqueness/non-uniqueness phase transition on the \Delta-regular tree. More generally, for any antiferromagnetic 2-spin system we prove O(n\log{n) mixing time of the Glauber dynamics on any bounded degree graph in the corresponding tree uniqueness region. Our results apply more broadly; for example, we also obtain O(nn) mixing for q-colorings of triangle-free graphs of maximum degree \Delta when the number of colors satisfies q > \alpha \Delta } where \alpha \approx 1.763, and O(mn) mixing for generating random matchings of any graph with bounded degree and m edges.
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\lambda<\lambda_c(\Delta) where \lambda_c(\Delta) is the critical point for the uniqueness/non-uniqueness phase transition on the \Delta-regular tree. More generally, for any antiferromagnetic 2-spin system we prove O(n\log{n
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\lambda<\lambda_c(\Delta) where \lambda_c(\Delta) is the critical point for the uniqueness/non-uniqueness phase transition on the \Delta-regular tree. More generally, for any antiferromagnetic 2-spin system we prove O(n\log{n