Abstract
The paper considers spin systems on the d-dimensional integer lattice ℤd with nearest-neighbor interactions. A sharp equivalence is proved between exponential decay with distance of spin correlations (a spatial property of the equilibrium state) and “super-fast” mixing time of the Glauber dynamics (a temporal property of a Markov chain Monte Carlo algorithm). While such an equivalence is already known in various forms, the proofs in this paper are purely combinatorial and avoid the functional analysis machinery employed in previous proofs.
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Supported by EPSRC grant “Sharper Analysis of Randomised Algorithms: a ComputationalApproach” and by EC IST project RAND-APX.
Supported in part by NSF grants CCR-9820951 and CCR-0121555, and by DARPA cooperative agreement F30602-00-2-060.
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Dyer, M., Sinclair, A., Vigoda, E., Weitz, D. (2002). Mixing in Time and Space for Lattice Spin Systems: A Combinatorial View. In: Rolim, J.D.P., Vadhan, S. (eds) Randomization and Approximation Techniques in Computer Science. RANDOM 2002. Lecture Notes in Computer Science, vol 2483. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45726-7_13
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DOI: https://doi.org/10.1007/3-540-45726-7_13
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