Date: Friday, January 26, 2018
Location: 4088 East Hall (4:10 PM to 5:00 PM)
Title: Double jump phase transition in a random soliton cellular automaton
Abstract: In this work, we consider the soliton cellular automaton introduced in cite{takahashi1990soliton} with a random initial configuration. We give multiple constructions of a Young diagram describing various statistics of the system in terms of familiar objects like birthanddeath chains and GaltonWatson forests. Using these ideas, we establish limit theorems showing that if the first $n$ boxes are occupied independently with probability $pin(0,1)$, then the number of solitons is of order $n$ for all $p$, and the length of the longest soliton is of order $log n$ for $p<1/2$, order $sqrt{n}$ for $p=1/2$, and order $n$ for $p>1/2$. Additionally, we uncover a condensation phenomenon in the supercritical regime: For each fixed $jgeq 1$, the top $j$ soliton lengths have the same order as the longest for $pleq 1/2$, whereas all but the longest have order at most $log n$ for $p>1/2$. As an application, we obtain scaling limits for the lengths of the $k^{ ext{th}}$ longest increasing and decreasing subsequences in a random stacksortable permutation of length $n$ in terms of random walks and Brownian excursions.
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Speaker: Hanbaek Lyu
Institution: Ohio State U.
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