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Central limit theorems for tableaux related to the partially asymmetric simple exclusion process
Dissertation   Open access

Central limit theorems for tableaux related to the partially asymmetric simple exclusion process

Aleksandr Yaroslavskiy
Doctor of Philosophy (Ph.D.), Drexel University
Jun 2020
DOI:
https://doi.org/10.17918/00001391
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Abstract

Combinatorial analysis Probabilities
In this thesis, we primarily study corners in Tree-like and Permutation Tableaux. Tree-like Tableaux are in bijection with other combinatorial structures, including Permutation Tableaux, and have a connection to the partially asymmetric simple exclusion process (PASEP), an important model of an interacting particles system. In particular, in the context of Tree-like Tableaux, a corner corresponds to a node occupied by a particle that could jump to the right while inner corners indicate a particle with an empty node to its left. Thus, the total number of corners represents the number of nodes at which PASEP can move, i.e., the total current activity of the system. As the number of inner corners and regular corners is connected, we limit our discussion to just regular corners and show that asymptotically, the number of corners in a tableau of length n is normally distributed. Furthermore, since the number of corners in Tree-like Tableaux are closely related to the number of corners in Permutation Tableaux, we discuss the corners in the context of the latter tableaux. Finally, using analogous techniques, we prove a central limit theorem for the number of corners in Symmetric Tree-like Tableaux and Type-B Permutation Tableaux.

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