Abstract We study the Brownian motion on the non-compact Grassmann manifold $\frac {\textbf {U}(n-k,k)} {\textbf {U}(n-k)\textbf {U}(k)}$ and some of its functionals. The key point is to realize this Brownian motion as a matrix diffusion process, use matrix stochastic calculus and take advantage of the hyperbolic Stiefel fibration to study a functional that can be understood in that setting as a generalized stochastic area process. In particular, a connection to the generalized Maass Laplacian of the complex hyperbolic space is presented and applications to the study of Brownian windings in the Lie group $\textbf {U}(n-k,k)$ are then given.
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Asymptotic windings of the block determinants of a unitary Brownian motion and related diffusions
We study several matrix diffusion processes constructed from a unitary Brownian motion. In particular, we use the Stiefel fibration to lift the Brownian motion of the complex Grassmannian to the complex Stiefel manifold and deduce a skew-product decomposition of the Stiefel Brownian motion. As an application, we prove asymptotic laws for the determinants of the block entries of the unitary Brownian motion.
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- NSF-PAR ID:
- 10221353
- Date Published:
- Journal Name:
- Electronic journal of probability
- Volume:
- 26
- Issue:
- 38
- ISSN:
- 1083-6489
- Page Range / eLocation ID:
- 1-21
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract The free multiplicative Brownian motion
$$b_{t}$$ N limit of the Brownian motion on$$\mathsf {GL}(N;\mathbb {C}),$$ $$*$$ N limit of the empirical distribution of eigenvalues is thus the Brown measure of$$b_{t}$$ $$\Sigma _{t}$$ $$W_{t}$$ $$\overline{\Sigma }_{t},$$ $$\Sigma _{t}$$ $$\begin{aligned} W_{t}(r,\theta )=\frac{1}{r^{2}}w_{t}(\theta ), \end{aligned}$$ $$w_{t}$$ $$\Sigma _{t}$$ $$u_{t}$$ $$b_{t}$$ -
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Abstract How fast a state of a system converges to a stationary state is one of the fundamental questions in science. Some Markov chains and random walks on finite groups are known to exhibit the non-asymptotic convergence to a stationary distribution, called the cutoff phenomenon. Here, we examine how quickly a random quantum circuit could transform a quantum state to a Haar-measure random quantum state. We find that random quantum states, as stationary states of random walks on a unitary group, are invariant under the quantum Fourier transform (QFT). Thus the entropic uncertainty of random quantum states has balanced Shannon entropies for the computational basis and the QFT basis. By calculating the Shannon entropy for random quantum states and the Wasserstein distances for the eigenvalues of random quantum circuits, we show that the cutoff phenomenon occurs for the random quantum circuit. It is also demonstrated that the Dyson-Brownian motion for the eigenvalues of a random unitary matrix as a continuous random walk exhibits the cutoff phenomenon. The results here imply that random quantum states could be generated with shallow random circuits.
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The particles in condensed matter physics are almost characterized by Brownian motion. This phenomenon is the basis for a very important understanding of the particles motion in condensed matter. For our previous research, there is already applied and confirmed the complex fractal correction which includes influence of parameters from grains and pores surface and also effects based on particles’ Brownian motion. As a chaotic structure of these motions, we have very complex research results regarding the particles’ trajectories in three-dimension (3D). In our research paper, we applied fractal interpolation within the idea to reconstruct the above mentioned trajectories in two dimensions at this stage. Because of the very complex fractional mathematics on Brownian motion, we found and developed much simpler and effective mathematization. The starting point is within linear interpolation. In our previous research, we presented very original line fractalization based on tensor product. But, in this paper, we applied and successfully confirmed that by fractal interpolation (Akimo polynomial method) that is possible to reconstruct the chaotical trajectories lines structures by several fractalized intervals and involved intervals. This novelty is very important because of the much more effective procedure that we can reconstruct and in that way control the particles’ trajectories. This is very important for further advanced research in microelectronics, especially inter-granular micro capacitors.more » « less
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Abstract The stochastic trajectories of molecules in living cells, as well as the dynamics in many other complex systems, often exhibit memory in their path over long periods of time. In addition, these systems can show dynamic heterogeneities due to which the motion changes along the trajectories. Such effects manifest themselves as spatiotemporal correlations. Despite the broad occurrence of heterogeneous complex systems in nature, their analysis is still quite poorly understood and tools to model them are largely missing. We contribute to tackling this problem by employing an integral representation of Mandelbrot’s fractional Brownian motion that is compliant with varying motion parameters while maintaining long memory. Two types of switching fractional Brownian motion are analysed, with transitions arising from a Markovian stochastic process and scale-free intermittent processes. We obtain simple formulas for classical statistics of the processes, namely the mean squared displacement and the power spectral density. Further, a method to identify switching fractional Brownian motion based on the distribution of displacements is described. A validation of the model is given for experimental measurements of the motion of quantum dots in the cytoplasm of live mammalian cells that were obtained by single-particle tracking.
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1214/21-EJP600
