stationary measure
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2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Luigi Cantini ◽  
Jan De Gier ◽  
Michael Wheeler

International audience We present a new, explicit sum formula for symmetric Macdonald polynomials Pλ and show that they can be written as a trace over a product of (infinite dimensional) matrices. These matrices satisfy the Zamolodchikov– Faddeev (ZF) algebra. We construct solutions of the ZF algebra from a rank-reduced version of the Yang–Baxter algebra. As a corollary, we find that the normalization of the stationary measure of the multi-species asymmetric exclusion process is a Macdonald polynomial with all variables set equal to one.



2019 ◽  
Vol 15 (4) ◽  
pp. 2805-2858
Author(s):  
Yves Benoist ◽  
Jean-François Quint
Keyword(s):  


Author(s):  
Krzysztof Barański ◽  
Adam Śpiewak

Abstract We show that the stationary measure for some random systems of two piecewise affine homeomorphisms of the interval is singular, verifying partially a conjecture by Alsedà and Misiurewicz and contributing to a question by Navas on the absolute continuity of stationary measures, considered in the setup of semigroups of piecewise affine circle homeomorphisms. We focus on the case of resonant boundary derivatives.





2019 ◽  
Vol 19 (11&12) ◽  
pp. 901-912
Author(s):  
Takako Endo ◽  
Takashi Komatsu ◽  
Norio Konno ◽  
Tomoyuki Terada

We focus on the three-state quantum walk (QW) in one dimension. In this paper, we give the stationary measure in general condition, originated from the eigenvalue problem. Firstly, we get the transfer matrices by our new recipe, and solve the eigenvalue problem. Then we obtain the general form of the stationary measure for concrete initial state and eigenvalue. We also show some specific examples of the stationary measure for the three-state QW. One of the interesting and crucial future problems is to make clear the whole picture of the set of stationary measures.



2019 ◽  
Vol 91 (3-4) ◽  
pp. 265-295
Author(s):  
Denis Denisov ◽  
Dmitry Korshunov ◽  
Vitali Wachtel


2018 ◽  
Vol 39 (12) ◽  
pp. 3207-3223
Author(s):  
YVES BENOIST ◽  
CAROLINE BRUÈRE

We study the action of the affine group $G$ of $\mathbb{R}^{d}$ on the space $X_{k,\,d}$ of $k$-dimensional affine subspaces. Given a compactly supported Zariski dense probability measure $\unicode[STIX]{x1D707}$ on $G$, we show that $X_{k,d}$ supports a $\unicode[STIX]{x1D707}$-stationary measure $\unicode[STIX]{x1D708}$ if and only if the $(k+1)\text{th}$ Lyapunov exponent of $\unicode[STIX]{x1D707}$ is strictly negative. In particular, when $\unicode[STIX]{x1D707}$ is symmetric, $\unicode[STIX]{x1D708}$ exists if and only if $2k\geq d$.



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