Determinantal Martingales and Correlations of Noncolliding Random Walks

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• 作者：Makoto Katori
• 关键词：Noncolliding random walk ; Discrete It?’s formula ; Martingales ; Determinantal processes ; Random matrix theory ; Infinite particle systems ; Invariance principle
• 刊名：Journal of Statistical Physics
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：159
• 期：1
• 页码：21-42
• 全文大小：332 KB
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• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Statistical Physics
  Mathematical and Computational Physics
  Physical Chemistry
  Quantum Physics
  
• 出版者：Springer Netherlands
• ISSN：1572-9613

文摘

We study the noncolliding random walk (RW), which is a particle system of one-dimensional, simple and symmetric RWs starting from distinct even sites and conditioned never to collide with each other. When the number of particles is finite, \(N , this discrete process is constructed as an \(h\) -transform of absorbing RW in the \(N\) -dimensional Weyl chamber. We consider Fujita’s polynomial martingales of RW with time-dependent coefficients and express them by introducing a complex Markov process. It is a complexification of RW, in which independent increments of its imaginary part are in the hyperbolic secant distribution, and it gives a discrete-time conformal martingale. The \(h\) -transform is represented by a determinant of the matrix, whose entries are all polynomial martingales. From this determinantal-martingale representation (DMR) of the process, we prove that the noncolliding RW is determinantal for any initial configuration with \(N , and determine the correlation kernel as a function of initial configuration. We show that noncolliding RWs started at infinite-particle configurations having equidistant spacing are well-defined as determinantal processes and give DMRs for them. Tracing the relaxation phenomena shown by these infinite-particle systems, we obtain a family of equilibrium processes parameterized by particle density, which are determinantal with the discrete analogues of the extended sine-kernel of Dyson’s Brownian motion model with \(\beta =2\) . Following Donsker’s invariance principle, convergence of noncolliding RWs to the Dyson model is also discussed.