Translator Disclaimer
September 2018 Zigzag diagrams and Martin boundary
Pierre Tarrago
Ann. Probab. 46(5): 2562-2620 (September 2018). DOI: 10.1214/17-AOP1234

Abstract

We investigate the asymptotic behavior of random paths on a graded graph which describes the subword order for words in two letters. This graph, denoted by $\mathcal{Z}$, has been introduced by Viennot, who also discovered a remarkable bijection between paths on $\mathcal{Z}$ and sequences of permutations. Later on, Gnedin and Olshanski used this bijection to describe the set of Gibbs measures on this graph. Both authors also conjectured that the Martin boundary of $\mathcal{Z}$ should coincide with its minimal boundary. We give here a proof of this conjecture by describing the distribution of a large random path conditioned on having a prescribed endpoint. We also relate paths on the graph $\mathcal{Z}$ with paths on the Young lattice, and we finally give a central limit theorem for the Plancherel measure on the set of paths in $\mathcal{Z}$.

Citation

Download Citation

Pierre Tarrago. "Zigzag diagrams and Martin boundary." Ann. Probab. 46 (5) 2562 - 2620, September 2018. https://doi.org/10.1214/17-AOP1234

Information

Received: 1 March 2015; Revised: 1 September 2017; Published: September 2018
First available in Project Euclid: 24 August 2018

zbMATH: 06964344
MathSciNet: MR3846834
Digital Object Identifier: 10.1214/17-AOP1234

Subjects:
Primary: 60C05, 60J45
Secondary: 05E99

Rights: Copyright © 2018 Institute of Mathematical Statistics

JOURNAL ARTICLE
59 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.46 • No. 5 • September 2018
Back to Top