Scaling limits and universality of Ising and dimer models

TL;DR

This paper reviews recent advances in understanding the universal scaling limits of two-dimensional Ising and dimer models at criticality, emphasizing robustness and conjectured structures of these limits.

Contribution

The paper presents new results and conjectures on the universality and structure of scaling limits for non-planar Ising models and interacting dimers, including proof strategies.

Findings

Progress in characterizing the limiting distribution of critical models

Evidence supporting the universality of scaling limits under perturbations

New conjectures on the structure of the scaling limit for non-solvable models

Abstract

After having introduced the notion of universality in statistical mechanics and its importance for our comprehension of the macroscopic behavior of interacting systems, I review recent progress in the understanding of the scaling limit of lattice critical models, including a quantitative characterization of the limiting distribution and the robustness of the limit under perturbations of the microscopic Hamiltonian. Specifically, I focus on two classes of non-exactly-solvable two-dimensional systems: non-planar Ising models and interacting dimers. In both settings, I describe the conjectures on the expected structure of the scaling limit, review the progress towards their proof, and state some of the recent results on the universality of the limit, which I contributed to. Finally, I outline the ideas and methods involved in the proofs, describe some of the perspectives opened by these results, and propose several open problems.