Statistical mechanics, structure of matter Disordered systems Theorem

Parisi formula

Builds on Sherrington–Kirkpatrick model Spin glass Probability measure

Statement

Let $\beta \ge 0$ and $h \in \R$ . For $N \ge 2$ , let $(g_{ij})_{1 \le i < j \le N}$ be independent standard Gaussian random variables, and let $F_N = \frac{1}{N} \mathbb{E} \log \sum_{\sigma \in \set{-1,+1}^N} \exp\Big( \frac{\beta}{\sqrt{N}} \sum_{i < j} g_{ij} \sigma_i \sigma_j + h \sum_{i=1}^N \sigma_i \Big)$ be the free energy of the Sherrington–Kirkpatrick model. For a probability measure $\zeta$ on $[0, 1]$ , let $\alpha(q) = \zeta([0, q])$ , and let $\Phi_\zeta \colon [0, 1] \times \R \to \R$ solve the Parisi equation $\partial_t \Phi_\zeta = -\frac{\beta^2}{2} \Big( \partial_x^2 \Phi_\zeta + \alpha(t) \, (\partial_x \Phi_\zeta)^2 \Big), \qquad \Phi_\zeta(1, x) = \log \cosh x.$ Define the Parisi functional $\mathcal{P}(\zeta) = \log 2 + \Phi_\zeta(0, h) - \frac{\beta^2}{2} \int_0^1 q \, \alpha(q) \, dq.$ Then $F_N$ converges as $N \to \infty$ , and $\lim_{N \to \infty} F_N = \inf_{\zeta} \mathcal{P}(\zeta),$ where the infimum runs over all probability measures $\zeta$ on $[0, 1]$ .

Remarks

For $\zeta$ with finite support, $\alpha$ is a step function, and the Parisi equation is solved step by step with the Cole–Hopf transform. The general case follows by continuity in $\zeta$ . Parisi found the formula around 1980 with the non-rigorous replica trick. Guerra proved in 2003 the upper bound $F_N \le \mathcal{P}(\zeta) + o(1)$ for every $\zeta$ by an interpolation argument. Talagrand proved the matching lower bound in 2006, which completed the proof.

Example: take $h = 0$ and $\zeta = \delta_0$ , so that $\alpha \equiv 1$ . The Cole–Hopf transform gives $\Phi_\zeta(0, x) = \log \mathbb{E} \cosh(x + \beta Z) = \log \cosh x + \beta^2 / 2$ with $Z$ standard Gaussian. Then $\mathcal{P}(\delta_0) = \log 2 + \beta^2/2 - \beta^2/4 = \log 2 + \beta^2/4$ , the annealed free energy. This value is the true limit for $\beta < 1$ , while for $\beta > 1$ the infimum is strictly smaller.

Going deeper

For a student

The infimum is attained at a unique measure $\zeta$ , the Parisi measure. In the physics picture, and in rigorous form for generic mixed $p$ -spin models, it is the limiting law of the overlap $\frac{1}{N} \sum_i \sigma^1_i \sigma^2_i$ of two independent samples from the Gibbs measure. At high temperature it is a single point mass, which is called replica symmetry. At low temperature it is expected to have a continuous part, which is called full replica symmetry breaking. Panchenko later extended the formula to mixed $p$ -spin models, with a proof based on the ultrametric structure of the overlaps. The same variational form appears in many problems on random constraint satisfaction and on high-dimensional statistics.

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Tag	`32S4Y36QNK`
URI	`m2://32S4Y36QNK`
Kind	statement, theorem
Writing	W0 stub
MSC	82B44, 60K35, 82D30
Status	This copy of the site has no build data. Run `m2 status 32S4Y36QNK`.
About	Sherrington–Kirkpatrick model
Uses	Sherrington–Kirkpatrick model, Spin glass, Probability measure
Changed	3 November 2026