Possible topics for postgraduate research

The two-periodic Aztec diamond and related models

The two-periodic Aztec diamond is a domino tiling of an Aztec diamond with a specific weighting for assigning tilings, the so-called two-periodic weights. Random tilings of two-periodic Aztec diamonds feature interesting features – three macroscopic phases emerge, known as frozen, rough and smooth phases, see the figure on the left for a simulation. In the frozen phase, the tiling is deterministic. In the rough phase, the correlations of tiles have polynomial decay and it is expected that the fluctuations of the height function are given by the Gaussian free field. The figure on the right shows the height function of a realization of a tiling of an Aztec diamond.

In the smooth phase, also known as the de-localized phase, the correlations of the tiles decay exponentially. The frozen-rough and rough-smooth phases exhibit fluctuations that were first observed in random matrix theory. Indeed, the extended Airy kernel point process is observed at these boundaries. The mathematics behind this model is extremely rich, with connections to algebraic combinatorics, representation theory, mathematical physics and probability theory. Projects around the two-periodic Aztec diamond will involve investigating the probabilistic, algebraic and combinatorial structures sitting behind this model as well as related models, which are expected to have similar behaviour.

For some references on previous research:

• A. Ayyer, S. Chhita and K. Johansson, GOE fluctuations for the maximum of the top path in alternating sign matrices, https://arxiv.org/abs/2109.02422
• S. Chhita and K. Johansson, Domino statistics of the two-periodic Aztec diamond. Adv. Math. 294 (2016), 37–149, https://arxiv.org/abs/1410.2385

For review papers and books:

• V. Gorin, Lectures on random lozenge tilings. Cambridge Stud. Adv. Math.Cambridge University Press, Cambridge, 2021
• I. Corwin, Kardar-Parisi-Zhang universality. Notices Amer. Math. Soc. 63 (2016), no. 3, 230–239

Contact: Sunil Chhita.