# Extreme unitary minimal models of conformal field theory

Some of the best understood conformal field theories are the 2D unitary minimal models $\mathcal{M}(m+1,m)$ indexed by the integer $m\ge 2$ and with central charge $$c=1-\frac{6}{m(m+1)}\ .$$ I would like to understand the extreme cases $m=2$ and $m=\infty$.

According to the article "Conformal symmetry and multicritical points in two-dimensional quantum field theory" by Zamolodchikov in Sov. J. Nucl. Phys. 1986 (the article is also reprinted in this book edited by Itzykson, Saleur and Zuber), these should be scale invariant cases of $\mathcal{P}(\phi)_2$ theories as studied in the constructive quantum field theory literature (e.g., in the classics by Glimm-Jaffe or Simon).

In this correspondence, $\mathcal{M}(m+1,m)$ is obtained by perturbing the Gaussian fixed point using a polynomial $\mathcal{P}(\phi)$ which is even, of degree $2m-2$, and with $m-1$ wells. Some of these minimal models have been identified as the scaling limits of specific statistical mechanics models: $m=3$ is the Ising model, $m=4$ is the tri-critical Ising model, $m=5$ is the three-state Potts model. However, almost all references I have seen say nothing about the simplest case $m=2$ which seems too degenerate to be worth mentioning.

Q1: Is $m=2$ really that trivial?

The multicritical correspondence suggests that this model is obtained by running the RG along a $\lambda\phi^2$ perturbation of the Gaussian fixed point. Equivalently, it would be a large-distance scaling limit of a massive free field with action $\frac{1}{2}\int\{(\partial\phi)^2+\lambda\phi^2$}. If this is the case then this limit should be 2D Gaussian white noise.

Q2: 2D Gaussian white noise is conformally invariant by a result of Hida and collaborators. Did anyone compute the central charge and show it is zero? Also, is there a relation between the $m=2$ minimal model and percolation (also with $c=0$) as well as black noise?

The other half of my question concerns the $m\rightarrow\infty$ limit. Then the multicritical correspondence suggests the field looks almost $\mathbb{Z}$-valued since the number of potential wells becomes infinite.

Q3: is there a good explanation of why this should look more and more like the massless free field with $c=1$?