06.06.2023, 15 Uhr
Phenomena in High Dimensions
Pierre Youssef (New York University, Abu Dhabi)
Béatrice De Tilière (IUF, University Paris Dauphine CEREMADE) and Sunil Chhita (Durham University)
2pm-2:45pm Béatrice De Tilière (IUF, University Paris Dauphine CEREMADE): The Z-invariant Ising model via dimers
3pm-3:45 pm Sunil Chhita (Durham University): The two-periodic Aztec diamond
Béatrice De Tilière (IUF, University Paris Dauphine CEREMADE): The Z-invariant Ising model via dimers
The Ising model belongs to the field of statistical mechanics and models ferromagnetism. Its Z-invariant version is defined on a planar, embedded graph satisfying a geometric constraint known as isoradiality, imposing that all faces can be inscribed in a circle of radius 1. Z-invariance imposes that the coupling constants satisfy the Ising model Yang-Baxter equations. The "classical" Ising models on the square, triangular or honeycomb lattice are specific examples of the above. The coupling constants are explicit, and depend on an elliptic parameter k playing the role of the temperature; in the specific case where k=0, the Ising model is critical.
In this talk we will define the model, and explain how it can be studied using dimers, also known as perfect matchings, of a related graph. We will then report on results obtained in collaboration with Cédric Boutillier (Sorbonne University) and Kilian Raschel (University of Tours). We will discuss the locality property of the Gibbs measure and of the free energy and establish that the model undergoes an order two phase transition at k=0, showing that this phase transition is the same as that of the rooted spanning forests model. This talk is aimed at a general audience; the models as well as the statistical mechanics terminology will be defined.
Sunil Chhita (Durham University): The two-periodic Aztec diamond
A random tiling of a bounded domain consists of taking some lattice domain and tiling it with elementary blocks, picking each tiling at random with some prescribed probability measure. In this talk, we focus on a few important results for random tiling models that have been discovered since the 1990's, using domino tilings of the Aztec diamond as a guide. Along the way, we introduce the two-periodic Aztec diamond, which is one of the few mathematically tractable models containing three macroscopic regions: frozen, where the dominoes are deterministic; rough, where the correlations between dominoes decay polynomially; smooth, where the correlations between dominoes decay exponentially. We present some results on the behavior at the rough-smooth interface as well as discussing the local geometry at this interface.
This talk is based on a series of joint works with Vincent Beffara, Kurt Johansson and Benjamin Young.
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