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Lectures on Glauber Dynamics for Discrete Spin Models

Lectures on Glauber Dynamics for Discrete Spin Models,10.1007/978-3-540-48115-7_2,Fabio Martinelli

Lectures on Glauber Dynamics for Discrete Spin Models   (Citations: 56)
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These notes have been the subject of a course I gave in the summer 1997 for the school in probability theory in Saint-Flour. I review in a self-contained way the state of the art, sometimes providing new and simpler proofs of the most relevant results, of the theory of Glauber dynamics for classical lattice spin models of statistical mechanics. The material covers the dynamics in the one phase region, in the presence of boundary phase transitions, in the phase coexistence region for the two dimensional Ising model and in the so-called Griffiths phase for random Systems.
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    • ...Supported by many experiments and studies in the theory of dynamical critical phenomena, physicists believe that the spectral-gap of the continuous-time dynamics on lattices has the following critical slowing down behavior (e.g., [16,20,25,38]): At high temperatures ( β< β c) the inverse-gap is O(1), at the critical βc it is polynomial in the surface-area and at low temperatures it is exponential in it. This is known for Z 2 except at ...
    • ...This provides an estimate of the spectral-gap of the single-site dynamics in terms of those of the individual blocks and the block-dynamics chain itself (see [25])...
    • ...This theorem appears in [25] in a more general setting, and following is its reformulation for the special case of Glauber dynamics for the Ising model on a finite graph with an arbitrary boundary condition...

    Jian Dinget al. Mixing Time of Critical Ising Model on Trees is Polynomial in the Heig...

    • ...[14,17] and the recent work on the cutoff phenomenon for the mean field Ising model [15])...
    • ...greater than δ> 0) as can be seen via standard comparison techniques [17]...

    Fabio Martinelliet al. On the Mixing Time of the 2D Stochastic Ising Model with “Plus” Bounda...

    • ...We use the following decomposition result of Lucier and Molloy [21], which is an application of the block dynamics technique (see, Proposition 3.4 in [23]) to the Glauber dynamics on the complete trees combined with Lemma 2 in Mossel and Sly [27]...

    Prasad Tetaliet al. Phase Transition for the Mixing Time of the Glauber Dynamics for Color...

    • ...L is defined with boundary-source choice (M, S) = (∅, ∂ − Q ˜ L ). A standard property known as finite speed of propagation (see [22]) asserts that for a proper C ,i fa is chosen large enough for all t,...

    N. Cancriniet al. Kinetically Constrained Lattice Gases

    • ...[17]. Much recent work has been devoted to determining sufficient and ne cessary conditions for rapid convergence of Gibbs samplers...
    • ...It is based on a combination of block dynamics, see e.g. [17], and path coupling, see e.g...
    • ...Proposition 1 If τblock is the relaxation time of the block dynamics andτi is the maximum the relaxation time on Vi given any boundary condition fromG − {Vi} then by Proposition 3.4 of [17]...
    • ...Applying Lemma 2 and 3 we get that the block dynamics satisfies τblock ≤ Ck. Then by Proposition 3.4 of [17] we have that...
    • ...for sufficiently large C. Then by Proposition 3.4 of [17] we have that...
    • ...the mixing time of the block dynamics is simply 1. By applying Proposition 3.4 of [17] we get that the relaxation time on T ′ is simply the maximum of the relaxation times on the blocks,...
    • ... Then by Proposition 3.4 of [17] we have that the relaxation time of the Glauber dynamics on Vj is bounded by nC . �...
    • ...The main results now follows easily using the block dynamics approach of Proposition 3.4 of [17]...
    • ... Then by Proposition 3.4 of [17] we have that the relaxation time is O(nC...

    Elchanan Mosselet al. Gibbs rapidly samples colorings of G(n, d/n)

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