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

Energy, disorder, and Gibbs law

Stage 0: freeze the classical model and the external claim › Energy, disorder, and Gibbs law · lines 209–251

Oai227-A@1#eq:energy

For $n \geq 2$ , put

Ω_{n} = {- 1, 1}^{n} .

For a symmetric array $J$ , with $J_{ii} = 0$ , define the classical energy

H_{n, J} (σ) = - \frac{1}{n} i < j \sum J_{ij} σ_{i} σ_{j} .

This is a real-valued function on a finite configuration space. It is not a quantum Hamiltonian.

At inverse temperature $β$ , the Gibbs law is

π_{n, J, β} (σ) = \frac{exp ( - β H _{n, J} ( σ ))}{\sum _{τ \in Ω_{n}} exp ( - β H _{n, J} ( τ ))} .

Throughout the local certificate,

β = 1, π = π_{n, J, 1} .

The two nominated disorder laws are

D_{G} = N (0, 1), D_{R} = \frac{1}{2} δ_{- 1} + \frac{1}{2} δ_{1},

independently on the edges $i < j$ . The normalized interaction matrix has off-diagonal entries $W_{ij} = J_{ij} / n$ .

These conventions are the zero-field SK normalization used in the external claim. The original SK model is classical SK1975,OAI227Quench.

Displayed formulas in this passage

Oai227-A@1#eq:energy · line 217
$H_{n, J} (σ) = - \frac{1}{n} i < j \sum J_{ij} σ_{i} σ_{j} .$
unlabeled · line 212
$Ω_{n} = {- 1, 1}^{n} .$
unlabeled · line 227
$π_{n, J, β} (σ) = \frac{exp ( - β H _{n, J} ( σ ))}{\sum _{τ \in Ω_{n}} exp ( - β H _{n, J} ( τ ))} .$
unlabeled · line 234
$β = 1, π = π_{n, J, 1} .$
unlabeled · line 239
$D_{G} = N (0, 1), D_{R} = \frac{1}{2} δ_{- 1} + \frac{1}{2} δ_{1},$