# nLab Blake Stacey

I’m a statistical-physics novice living and working in the general vicinity of Boston, Massachusetts. I hang around the nLab to keep my intellectual curiosity engaged, and sometimes I contribute a bibliographic item or a copy-edit.

Evidence of my lack of education:

On the advice of John Baez, I’m going to try writing a bit about universality. And having started a page on universality classes, I now realise that I should get a few other topics going:

• c-theorem
• Reggeon field theory?
• RSOS model?
• ADE classification?

### Scratchpad

Restricted Solid-on-Solid or RSOS models are field theories defined on a lattice, in which the field values at adjacent lattice sites are energetically constrained in a way specified by a graph structure. If the graph is one of the special $A$, $D$ or $E$ Dynkin diagrams, then the critical point of the model is governed by a conformal field theory whose central charge is fixed by the Coxeter number of that Dynkin diagram. RSOS models therefore exemplify the possible universality classes of 2D systems in thermal equilibrium.

$Z=\sum _{\mathrm{heights}}\prod _{\mathrm{faces}}W\left(\begin{array}{cc}d& c\\ a& b\end{array}\right).$Z = \sum_{heights} \prod_{faces} W \left(\begin{array}{cc} d & c \\ a & b\end{array}\right).

The Boltzmann weights, as defined by Pasquier, depend on the crossing parameter $\lambda =\pi /\stackrel{^}{h}$, where $\stackrel{^}{h}$ is the Coxeter number of the Dynkin diagram, and the spectral parameter $u\in \left(0,\lambda \right)$.

$W\left(\begin{array}{cc}d& c\\ a& b\end{array}\mid u\right)=\frac{\mathrm{sin}\lambda -u}{\mathrm{sin}\lambda }{\delta }_{a,c}{G}_{a,b}{G}_{a,d}+\frac{\mathrm{sin}u}{\mathrm{sin}\lambda }{\left(\frac{{S}_{a}{S}_{c}}{{S}_{b}{S}_{d}}\right)}^{1/2}{\delta }_{b,d}{G}_{a,b}{G}_{b,c}.$W \left(\begin{array}{cc} d & c \\ a & b\end{array}\mid u\right) = \frac{\sin\lambda - u}{\sin\lambda} \delta_{a, c} G_{a, b} G_{a, d} + \frac{\sin u}{\sin\lambda} \left(\frac{S_a S_c}{S_b S_d}\right)^{1/2} \delta_{b, d} G_{a, b} G_{b, c}.

${G}_{a,b}$ is the adjacency matrix of the graph.

Michael E. Fisher (1968), Renormalization of Critical Exponents by Hidden Variables Physical Review 176: 257–72. (web)

The mathematical content of this article consists chiefly in the analysis of some thermodynamic manipulations appropriate to the imposition of “constraints” on certain previously free thermodynamic variables and is quite elementary. To motivate our analysis and reveal its theoretical significance we ask the following fundamental question: “To what extent are the observed values of the critical-point exponents universal?” … the fairly close concurrence of all the exponent values for fluids and magnets does suggest the existence of a universal set of exponents.

category: people

Revised on September 24, 2011 22:52:17 by Blake Stacey (209.6.90.149)