Manuscript Oai227-A@1 · Oai227-A.tex · section · 7f45f607

Heat-bath clocks and generator

Stage 0: freeze the classical model and the external claim › Heat-bath clocks and generator · lines 252–285

Oai227-A@1#eq:site, Oai227-A@1#eq:generator

Write $σ^{i, a}$ for the configuration obtained by setting coordinate $i$ to $a$ . At criticality the local field is

h_{i} (σ) = \frac{1}{n} j \neq = i \sum J_{ij} σ_{j} .

The site heat-bath operator is

(K_{i} f) (σ) = a = \pm 1 \sum \frac{e ^{a h_{i} (σ)}}{2 cosh h _{i} ( σ )} f (σ^{i, a}) .

Every site has an independent rate-one Poisson clock. At a ring its spin is sampled from (eq:site), including the possibility that it keeps its value.

The generator and semigroup are

L_{J} = i = 1 \sum^{n} (K_{i} - I), P_{u}^{J} = e^{u L_{J}} .

Thus one physical time unit contains $n$ update attempts on average.

Let $U_{n}$ denote the uniform law on $Ω_{n}$ , independent of $J$ . This is the quench initialization. It is not replaced by a Gibbs sample.

Displayed formulas in this passage

Oai227-A@1#eq:site · line 260
$(K_{i} f) (σ) = a = \pm 1 \sum \frac{e ^{a h_{i} (σ)}}{2 cosh h _{i} ( σ )} f (σ^{i, a}) .$
Oai227-A@1#eq:generator · line 273
$L_{J} = i = 1 \sum^{n} (K_{i} - I), P_{u}^{J} = e^{u L_{J}} .$
unlabeled · line 256
$h_{i} (σ) = \frac{1}{n} j \neq = i \sum J_{ij} σ_{j} .$

Heat-bath clocks and generator

Displayed formulas in this passage

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