Abstract
In these notes, we discuss a selection of topics on several models of planar statistical mechanics. We consider the Ising, Potts, and more generally abelian spin models; the discrete Gaussian free field; the random cluster model; and the six-vertex model. Emphasis is put on duality, order, disorder and spinor variables, and on mappings between these models.
Highlights
In the type of statistical models considered here, a fixed underlying graph carries a random realization of a structure, consisting of values carried by bonds or sites, and usually interacting via local rules
We will focus on the Ising and Potts models, where each site carries a random “spin”; the discrete Gaussian free field, where a scalar height function is defined on sites; the random cluster model, where realizations are random subgraphs; and the six-vertex model, where each edge has a random orientation
The key question is to understand the large scale random structures generated by these local interactions
Summary
In the type of statistical models considered here, a fixed underlying graph carries a random realization of a structure, consisting of values (boolean, in a finite alphabet or scalar) carried by bonds (edges) or sites (vertices), and usually interacting via local rules. Of particular interest are the “fixed points” of these duality transformations; they often coincide with a critical point of the model, at which one can observe large scale random structures. The field-theoretic concept of fermion finds a combinatorial incarnation as the combination of an order and a disorder variable at microscopic distance. Such combinations are referred to as parafermionic or spinor variables. In these notes, we will describe duality, order, disorder, and spinor variables for abelian spin models and the discrete free field. These include: phase transition, transfer matrices and Bethe ansatz [2, 38], Yang-Baxter solvability [2], infinite volume measures [15, 18] and limit shapes [3, 27, 38], scaling limits and Schramm-Loewner Evolution [47, 43]
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