Embedded Files
Disclaimer: Code for generating images of the Aztec diamond provided by Christophe Charlier through his website. Plots of the KdV and Toda solution were produced with the help of AI.
KdV_Simulation.mp4Typical numerically computed solution to the KdV equation with smooth steplike initial data. Observed the modulated wave region emerging from the initial step (see [2], [5]).
Toda_3Soliton_FlaschkaV2.mp4A three-soliton solution of the Toda lattice equation. Observe that the solitons retain their form after their nonlinear interaction. They also undergo a phase shift, but it is difficult to see as it is rather small.
Uniform tiling of the Aztec diamond displaying the arctic circle theorem, see [3].
Tiling with i.i.d. standard lognormal weights, conjectured to display a super-rough region, see [4].
Graph of the generalized discriminant (see [6]) associated to the 2x2-periodic model studied in [1]. Its zero-set is also known as the octic circle, as it is a degree 8 curve.
Literature
Literature
[1] S. Chhita and K. Johansson, Domino statistics of the two-periodic Aztec diamond, Adv. Math. 294, 37-149 (2016), doi.org/10.1016/j.aim.2016.02.025
[2] I. Egorova, M. Piorkowski and G. Teschl, Asymptotics of the Korteweg–de Vries shock waves via the Riemann–Hilbert approach, Indiana Univ. Math. J., 73 No. 2, 645-690 (2024),
[3] W. Jockusch, J. Propp and P. Shor, Random Domino Tilings and the Arctic Circle Theorem, arXiv:math/9801068
[4] A. Perret, Z. Ristivojevic, P. Le Doussal, G. Schehr and K.J. Wiese, Super Rough Glassy Phase of the Random Field XY Model in Two Dimensions, Phys. Rev. Lett. 109, 157205 (2012), doi.org/10.1103/PhysRevLett.109.157205
[5] M. Piorkowski, Parametrix problem for the Korteweg–de Vries equation with steplike initial data, J. of Differential Equations, 375, 280–314 (2023), doi.org/10.1016/j.jde.2023.06.052
[6] M. Piorkowski, Arctic curves of periodic dimer models and generalized discriminants, arXiv:2410.17138 (slides)
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