Integrability of Box-Ball Systems and Randomized Box-Ball Systems - Part 1

Explore the integrability of box-ball systems and randomized box-ball systems in this comprehensive 55-minute lecture by Atsuo Kuniba. Delve into the emergence of box-ball systems from classical integrable systems through ultradiscretization and as quantum integrable systems at q = 0. Examine the synthesis of these dual origins of integrability, focusing on combinatorial Bethe ansatz and crystals theory in quantum groups. Discover how these concepts relate to soliton theory, including rigged configurations as action-angle variables, KKR bijection as inverse scattering transformation, and fermionic character formula as partition function. Learn about the recently formulated complete box ball system in higher rank with a completely diagonal S matrix. Additionally, investigate randomized box-ball systems, covering topics such as density plateaux, current correlations, and large deviations. Gain insights into analytical and simulation results obtained through Thermodynamic Bethe Ansatz (TBA) and Generalized Hydrodynamics (GHD), including exact solutions of TBA equations, limit shapes of conserved Young diagrams, and long-time behavior of generalized current correlations.