Probability Seminar

Hanbaek LyuUniversity of Wisconsin, Madison
Scaling limit of soliton lengths in a multicolor box-ball system

Monday, April 10, 2023 - 3:45pm
Malott 406

Abstract: The box-ball systems are integrable cellular automata whose long-time behavior is characterized by soliton solutions, with rich connections to other integrable systems such as the Korteweg-de Vries equation. In this talk, we consider a multicolor box-ball system with two types of random initial configurations and obtain sharp scaling limits of the soliton lengths as the system size tends to infinity. We obtain sharp scaling limit of soliton lengths that turns out to be different from the single color case as established in Levine, Lyu, and Pike ’20. A large part of our analysis is devoted to study the associated carrier process, which is a multi-dimensional Markov chain on the orthant, whose excursions and running maxima are closely related to soliton lengths. We establish the sharp scaling of its ruin probabilities, Skorokhod decomposition, strong law of large numbers, and weak diffusive scaling limit to a semimartingale reflecting Brownian motion with explicit parameters. We also establish and utilize complementary descriptions of the soliton lengths and numbers in terms of the modified Greene-Kleitman invariants for the box-ball systems and associated circular exclusion processes.