Spectral Independence Beyond Uniqueness using the topological method
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We present novel results for fast mixing of Glauber dynamics using the newly introduced and powerful Spectral Independence method from [Anari, Liu, Oveis-Gharan: FOCS 2020]. In our results, the parameters of the Gibbs distribution are expressed in terms of the spectral radius of the adjacency matrix of G, or that of the Hashimoto non-backtracking matrix. The analysis relies on new techniques that we introduce to bound the maximum eigenvalue of the pairwise influence matrix ℐ^Λ,τ_G for the two spin Gibbs distribution μ. There is a common framework that underlies these techniques which we call the topological method. The idea is to systematically exploit the well-known connections between ℐ^Λ,τ_G and the topological construction called tree of self-avoiding walks. Our approach is novel and gives new insights to the problem of establishing spectral independence for Gibbs distributions. More importantly, it allows us to derive new – improved – rapid mixing bounds for Glauber dynamics on distributions such as the Hard-core model and the Ising model for graphs that the spectral radius is smaller than the maximum degree.
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