Spectral theory for the q-Boson particle system
Spectral theory for the q-Boson particle system
复制标题
DOI：
10.1112/s0010437x14007532
复制
发表时间：
2015-01-01
影响因子：
1.8
通讯作者：
Sasamoto, Tomohiro
中科院分区：
文献类型：
--
作者：
Borodin, Alexei;Corwin, Ivan;Sasamoto, Tomohiro
我们开发的q-玻色子（随机）粒子系统的生成器的谱理论。我们的中心结果是这个系统的Plancherel型同构定理。这个定理有各种含义。证明了q-玻色子生成元的Bethe变换的完备性，从而使我们能够求解一般初始数据的Kolmogorov前向和后向方程。由于q-TASEP（q-变形完全不对称简单排斥过程）的Markov对偶性，从一般的初始条件出发，得到了刻画q-TASEP定时分布的矩公式.该定理还蕴涵了左、右本征函数的双正交性。我们考虑我们的q-玻色子结果的限制，以前考虑的离散三角洲玻色气体货车Diejen，以及另一个离散三角洲玻色气体，描述了半离散随机热方程（或等效的，O 'Connell-Yor半离散定向聚合物配分函数）的时刻的演变。进一步的限制把我们带到三角洲玻色气体中产生的随机热方程/Kardar-Parisi-Zhang方程的研究时刻。
We develop spectral theory for the generator of the q-Boson (stochastic) particle system. Our central result is a Plancherel type isomorphism theorem for this system. This theorem has various implications. It proves the completeness of the Bethe ansatz for the q-Boson generator and consequently enables us to solve the Kolmogorov forward and backward equations for general initial data. Owing to a Markov duality with q-TASEP (q-deformed totally asymmetric simple exclusion process), this leads to moment formulas which characterize the fixed time distribution of q-TASEP started from general initial conditions. The theorem also implies the biorthogonality of the left and right eigenfunctions. We consider limits of our q-Boson results to a discrete delta Bose gas considered previously by van Diejen, as well as to another discrete delta Bose gas that describes the evolution of moments of the semi-discrete stochastic heat equation (or equivalently, the O'Connell-Yor semi-discrete directed polymer partition function). A further limit takes us to the delta Bose gas which arises in studying moments of the stochastic heat equation/Kardar-Parisi-Zhang equation.