Coulomb gas method

Definition

Let $a > 0$ be a hard-core radius, $K > 0$ a coupling and $z > 0$ a fugacity. Let $D \subset \R^2$ be a bounded region. A configuration of the two-dimensional Coulomb gas in $D$ is a family of points $x_1, \dots, x_m \in D$ with charges $q_1, \dots, q_m \in \set{-1, +1}$. It is required that $\sum_i q_i = 0$ and $\abs{x_i - x_j} \ge a$ for $i \ne j$. The energy of the configuration is $$E = -K \sum_{1 \le i < j \le m} q_i q_j \log \frac{\abs{x_i - x_j}}{a}.$$ The grand partition function of the gas is $$Z = \sum_{m \ge 0} \frac{z^m}{(m/2)!^2} \int e^{-E} \, dx_1 \cdots dx_m.$$ Here $m$ runs over even numbers, and the charges are $q_1 = \cdots = q_{m/2} = +1$ and $q_{m/2 + 1} = \cdots = q_m = -1$. The integral runs over admissible positions, and the factorials remove the labelling of charges of equal sign. The Coulomb gas method applies to a two-dimensional lattice model. It writes the partition function or the correlation functions of the model as those of such a Coulomb gas, exactly or up to irrelevant terms. Equivalently, it writes them as those of a Gaussian free field with point charges. It then derives the critical behaviour of the model from this Gaussian theory.