74 research outputs found

    Glauber Dynamics for the mean-field Potts Model

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    We study Glauber dynamics for the mean-field (Curie-Weiss) Potts model with q3q\geq 3 states and show that it undergoes a critical slowdown at an inverse-temperature βs(q)\beta_s(q) strictly lower than the critical βc(q)\beta_c(q) for uniqueness of the thermodynamic limit. The dynamical critical βs(q)\beta_s(q) is the spinodal point marking the onset of metastability. We prove that when β<βs(q)\beta<\beta_s(q) the mixing time is asymptotically C(β,q)nlognC(\beta, q) n \log n and the dynamics exhibits the cutoff phenomena, a sharp transition in mixing, with a window of order nn. At β=βs(q)\beta=\beta_s(q) the dynamics no longer exhibits cutoff and its mixing obeys a power-law of order n4/3n^{4/3}. For β>βs(q)\beta>\beta_s(q) the mixing time is exponentially large in nn. Furthermore, as ββs\beta \uparrow \beta_s with nn, the mixing time interpolates smoothly from subcritical to critical behavior, with the latter reached at a scaling window of O(n2/3)O(n^{-2/3}) around βs\beta_s. These results form the first complete analysis of mixing around the critical dynamical temperature --- including the critical power law --- for a model with a first order phase transition.Comment: 45 pages, 5 figure
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