47 research outputs found
Scaling limit of vicious walks and two-matrix model
We consider the diffusion scaling limit of the one-dimensional vicious walker
model of Fisher and derive a system of nonintersecting Brownian motions. The
spatial distribution of particles is studied and it is described by use of
the probability density function of eigenvalues of Gaussian random
matrices. The particle distribution depends on the ratio of the observation
time and the time interval in which the nonintersecting condition is
imposed. As is going on from 0 to 1, there occurs a transition of
distribution, which is identified with the transition observed in the
two-matrix model of Pandey and Mehta. Despite of the absence of matrix
structure in the original vicious walker model, in the diffusion scaling limit,
accumulation of contact repulsive interactions realizes the correlated
distribution of eigenvalues in the multimatrix model as the particle
distribution.Comment: REVTeX4, 12 pages, no figure, minor corrections made for publicatio
Moments of vicious walkers and M\"obius graph expansions
A system of Brownian motions in one-dimension all started from the origin and
conditioned never to collide with each other in a given finite time-interval
is studied. The spatial distribution of such vicious walkers can be
described by using the repulsive eigenvalue-statistics of random Hermitian
matrices and it was shown that the present vicious walker model exhibits a
transition from the Gaussian unitary ensemble (GUE) statistics to the Gaussian
orthogonal ensemble (GOE) statistics as the time is going on from 0 to .
In the present paper, we characterize this GUE-to-GOE transition by presenting
the graphical expansion formula for the moments of positions of vicious
walkers. In the GUE limit , only the ribbon graphs contribute and the
problem is reduced to the classification of orientable surfaces by genus.
Following the time evolution of the vicious walkers, however, the graphs with
twisted ribbons, called M\"obius graphs, increase their contribution to our
expansion formula, and we have to deal with the topology of non-orientable
surfaces. Application of the recent exact result of dynamical correlation
functions yields closed expressions for the coefficients in the M\"obius
expansion using the Stirling numbers of the first kind.Comment: REVTeX4, 11 pages, 1 figure. v.2: calculations of the Green function
and references added. v.3: minor additions and corrections made for
publication in Phys.Rev.
Universality for orthogonal and symplectic Laguerre-type ensembles
We give a proof of the Universality Conjecture for orthogonal (beta=1) and
symplectic (beta=4) random matrix ensembles of Laguerre-type in the bulk of the
spectrum as well as at the hard and soft spectral edges. Our results are stated
precisely in the Introduction (Theorems 1.1, 1.4, 1.6 and Corollaries 1.2, 1.5,
1.7). They concern the appropriately rescaled kernels K_{n,beta}, correlation
and cluster functions, gap probabilities and the distributions of the largest
and smallest eigenvalues. Corresponding results for unitary (beta=2)
Laguerre-type ensembles have been proved by the fourth author in [23]. The
varying weight case at the hard spectral edge was analyzed in [13] for beta=2:
In this paper we do not consider varying weights.
Our proof follows closely the work of the first two authors who showed in
[7], [8] analogous results for Hermite-type ensembles. As in [7], [8] we use
the version of the orthogonal polynomial method presented in [25], [22] to
analyze the local eigenvalue statistics. The necessary asymptotic information
on the Laguerre-type orthogonal polynomials is taken from [23].Comment: 75 page
Massive chiral random matrix ensembles at beta = 1 & 4 : QCD Dirac operator spectra
The zero momentum sectors in effective theories of QCD coupled to pseudoreal
(two colors) and real (adjoint) quarks have alternative descriptions in terms
of chiral orthogonal and symplectic ensembles of random matrices. Using this
correspondence, we compute correlation functions of Dirac operator eigenvalues
within a sector with an arbitrary topological charge in a presence of finite
quark masses of the order of the smallest Dirac eigenvalue. These novel
correlation functions, expressed in terms of Pfaffians, interpolate between
known results for the chiral and quenched limits as quark masses vary.Comment: 12 pages, REVTeX 3.1, 2 figures; (v2) corrections on signatures in
eqs.(48), (53), (C7), and on referential note
Determinantal process starting from an orthogonal symmetry is a Pfaffian process
When the number of particles is finite, the noncolliding Brownian motion
(BM) and the noncolliding squared Bessel process with index
(BESQ) are determinantal processes for arbitrary fixed initial
configurations. In the present paper we prove that, if initial configurations
are distributed with orthogonal symmetry, they are Pfaffian processes in the
sense that any multitime correlation functions are expressed by Pfaffians. The
skew-symmetric matrix-valued correlation kernels of the Pfaffians
processes are explicitly obtained by the equivalence between the noncolliding
BM and an appropriate dilatation of a time reversal of the temporally
inhomogeneous version of noncolliding BM with finite duration in which all
particles start from the origin, , and by the equivalence between
the noncolliding BESQ and that of the noncolliding squared
generalized meander starting from .Comment: v2: AMS-LaTeX, 17 pages, no figure, corrections made for publication
in J.Stat.Phy
Distribution of the k-th smallest Dirac operator eigenvalue
Based on the exact relationship to Random Matrix Theory, we derive the
probability distribution of the k-th smallest Dirac operator eigenvalue in the
microscopic finite-volume scaling regime of QCD and related gauge theories.Comment: REVTeX 3.1, 6 pages, 1 figure. Corrected factors in eqs. (16a) and
(16c) and very minor typos (v2
Massive random matrix ensembles at beta = 1 & 4 : QCD in three dimensions
The zero momentum sectors in effective theories of three dimensional QCD
coupled to pseudoreal (two colors) and real (adjoint) quarks in a classically
parity-invariant manner have alternative descriptions in terms of orthogonal
and symplectic ensembles of random matrices. Using this correspondence, we
compute finite-volume QCD partition functions and correlation functions of
Dirac operator eigenvalues in a presence of finite quark masses of the order of
the smallest Dirac eigenvalue. These novel correlation functions, expressed in
terms of quaternion determinants, are reduced to conventional results for the
Gaussian ensembles in the quenched limit.Comment: 26 pages, REVTeX 3.1 + mathlett.sty (attached), 4 figure
Microscopic universality with dynamical fermions
It has recently been demonstrated in quenched lattice simulations that the
distribution of the low-lying eigenvalues of the QCD Dirac operator is
universal and described by random-matrix theory. We present first evidence that
this universality continues to hold in the presence of dynamical quarks. Data
from a lattice simulation with gauge group SU(2) and dynamical staggered
fermions are compared to the predictions of the chiral symplectic ensemble of
random-matrix theory with massive dynamical quarks. Good agreement is found in
this exploratory study. We also discuss implications of our results.Comment: 5 pages, 3 figures, minor modifications, to appear in Phys. Rev. D
(Rapid Commun.
Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems
As an extension of the theory of Dyson's Brownian motion models for the
standard Gaussian random-matrix ensembles, we report a systematic study of
hermitian matrix-valued processes and their eigenvalue processes associated
with the chiral and nonstandard random-matrix ensembles. In addition to the
noncolliding Brownian motions, we introduce a one-parameter family of
temporally homogeneous noncolliding systems of the Bessel processes and a
two-parameter family of temporally inhomogeneous noncolliding systems of Yor's
generalized meanders and show that all of the ten classes of eigenvalue
statistics in the Altland-Zirnbauer classification are realized as particle
distributions in the special cases of these diffusion particle systems. As a
corollary of each equivalence in distribution of a temporally inhomogeneous
eigenvalue process and a noncolliding diffusion process, a stochastic-calculus
proof of a version of the Harish-Chandra (Itzykson-Zuber) formula of integral
over unitary group is established.Comment: LaTeX, 27 pages, 4 figures, v3: Minor corrections made for
publication in J. Math. Phy