25,258 research outputs found
Central Limit Theorems for the Brownian motion on large unitary groups
In this paper, we are concerned with the large N limit of linear combinations
of the entries of a Brownian motion on the group of N by N unitary matrices. We
prove that the process of such a linear combination converges to a Gaussian
one. Various scales of time and various initial distribution are concerned,
giving rise to various limit processes, related to the geometric construction
of the unitary Brownian motion. As an application, we propose a quite short
proof of the asymptotic Gaussian feature of the linear combinations of the
entries of Haar distributed random unitary matrices, a result already proved by
Diaconis et al.Comment: 14 page
Semiclassical Approach to Parametric Spectral Correlation with Spin 1/2
The spectral correlation of a chaotic system with spin 1/2 is universally
described by the GSE (Gaussian Symplectic Ensemble) of random matrices in the
semiclassical limit. In semiclassical theory, the spectral form factor is
expressed in terms of the periodic orbits and the spin state is simulated by
the uniform distribution on a sphere. In this paper, instead of the uniform
distribution, we introduce Brownian motion on a sphere to yield the parametric
motion of the energy levels. As a result, the small time expansion of the form
factor is obtained and found to be in agreement with the prediction of
parametric random matrices in the transition within the GSE universality class.
Moreover, by starting the Brownian motion from a point distribution on the
sphere, we gradually increase the effect of the spin and calculate the form
factor describing the transition from the GOE (Gaussian Orthogonal Ensemble)
class to the GSE class.Comment: 25 pages, 2 figure
Fixed energy universality for generalized Wigner matrices
We prove the Wigner-Dyson-Mehta conjecture at fixed energy in the bulk of the
spectrum for generalized symmetric and Hermitian Wigner matrices. Previous
results concerning the universality of random matrices either require an
averaging in the energy parameter or they hold only for Hermitian matrices if
the energy parameter is fixed. We develop a homogenization theory of the Dyson
Brownian motion and show that microscopic universality follows from mesoscopic
statistics
Dynamical Correlations among Vicious Random Walkers
Nonintersecting motion of Brownian particles in one dimension is studied. The
system is constructed as the diffusion scaling limit of Fisher's vicious random
walk. N particles start from the origin at time t=0 and then undergo mutually
avoiding Brownian motion until a finite time t=T. In the short time limit , the particle distribution is asymptotically described by Gaussian
Unitary Ensemble (GUE) of random matrices. At the end time t = T, it is
identical to that of Gaussian Orthogonal Ensemble (GOE). The Brownian motion is
generally described by the dynamical correlations among particles at many times
between t=0 and t=T. We show that the most general dynamical
correlations among arbitrary number of particles at arbitrary number of times
are written in the forms of quaternion determinants. Asymptotic forms of the
correlations in the limit are evaluated and a discontinuous
transition of the universality class from GUE to GOE is observed.Comment: REVTeX3.1, 4 pages, no figur
Non-intersecting Brownian walkers and Yang-Mills theory on the sphere
We study a system of N non-intersecting Brownian motions on a line segment
[0,L] with periodic, absorbing and reflecting boundary conditions. We show that
the normalized reunion probabilities of these Brownian motions in the three
models can be mapped to the partition function of two-dimensional continuum
Yang-Mills theory on a sphere respectively with gauge groups U(N), Sp(2N) and
SO(2N). Consequently, we show that in each of these Brownian motion models, as
one varies the system size L, a third order phase transition occurs at a
critical value L=L_c(N)\sim \sqrt{N} in the large N limit. Close to the
critical point, the reunion probability, properly centered and scaled, is
identical to the Tracy-Widom distribution describing the probability
distribution of the largest eigenvalue of a random matrix. For the periodic
case we obtain the Tracy-Widom distribution corresponding to the GUE random
matrices, while for the absorbing and reflecting cases we get the Tracy-Widom
distribution corresponding to GOE random matrices. In the absorbing case, the
reunion probability is also identified as the maximal height of N
non-intersecting Brownian excursions ("watermelons" with a wall) whose
distribution in the asymptotic scaling limit is then described by GOE
Tracy-Widom law. In addition, large deviation formulas for the maximum height
are also computed.Comment: 37 pages, 4 figures, revised and published version. A typo has been
corrected in Eq. (10
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