2,766 research outputs found
From Random Matrices to Stochastic Operators
We propose that classical random matrix models are properly viewed as finite
difference schemes for stochastic differential operators. Three particular
stochastic operators commonly arise, each associated with a familiar class of
local eigenvalue behavior. The stochastic Airy operator displays soft edge
behavior, associated with the Airy kernel. The stochastic Bessel operator
displays hard edge behavior, associated with the Bessel kernel. The article
concludes with suggestions for a stochastic sine operator, which would display
bulk behavior, associated with the sine kernel.Comment: 41 pages, 5 figures. Submitted to Journal of Statistical Physics.
Changes in this revision: recomputed Monte Carlo simulations, added reference
[19], fit into margins, performed minor editin
Airy processes and variational problems
We review the Airy processes; their formulation and how they are conjectured
to govern the large time, large distance spatial fluctuations of one
dimensional random growth models. We also describe formulas which express the
probabilities that they lie below a given curve as Fredholm determinants of
certain boundary value operators, and the several applications of these
formulas to variational problems involving Airy processes that arise in
physical problems, as well as to their local behaviour.Comment: Minor corrections. 41 pages, 4 figures. To appear as chapter in "PASI
Proceedings: Topics in percolative and disordered systems
{\bf -Function Evaluation of Gap Probabilities in Orthogonal and Symplectic Matrix Ensembles}
It has recently been emphasized that all known exact evaluations of gap
probabilities for classical unitary matrix ensembles are in fact
-functions for certain Painlev\'e systems. We show that all exact
evaluations of gap probabilities for classical orthogonal matrix ensembles,
either known or derivable from the existing literature, are likewise
-functions for certain Painlev\'e systems. In the case of symplectic
matrix ensembles all exact evaluations, either known or derivable from the
existing literature, are identified as the mean of two -functions, both
of which correspond to Hamiltonians satisfying the same differential equation,
differing only in the boundary condition. Furthermore the product of these two
-functions gives the gap probability in the corresponding unitary
symmetry case, while one of those -functions is the gap probability in
the corresponding orthogonal symmetry case.Comment: AMS-Late
Control of Integrable Hamiltonian Systems and Degenerate Bifurcations
We discuss control of low-dimensional systems which, when uncontrolled, are
integrable in the Hamiltonian sense. The controller targets an exact solution
of the system in a region where the uncontrolled dynamics has invariant tori.
Both dissipative and conservative controllers are considered. We show that the
shear flow structure of the undriven system causes a Takens-Bogdanov
birfurcation to occur when control is applied. This implies extreme noise
sensitivity. We then consider an example of these results using the driven
nonlinear Schrodinger equation.Comment: 25 pages, 11 figures, resubmitted to Physical Review E March 2004
(originally submitted June 2003), added content and reference
Variance Calculations and the Bessel Kernel
In the Laguerre ensemble of N x N (positive) hermitian matrices, it is of
interest both theoretically and for applications to quantum transport problems
to compute the variance of a linear statistic, denoted var_N f, as N->infinity.
Furthermore, this statistic often contains an additional parameter alpha for
which the limit alpha->infinity is most interesting and most difficult to
compute numerically. We derive exact expressions for both lim_{N->infinity}
var_N f and lim_{alpha->infinity}lim_{N->infinity} var_N f.Comment: 7 pages; resubmitted to make postscript compatibl
Edgeworth Expansion of the Largest Eigenvalue Distribution Function of GUE Revisited
We derive expansions of the resolvent
Rn(x;y;t)=(Qn(x;t)Pn(y;t)-Qn(y;t)Pn(x;t))/(x-y) of the Hermite kernel Kn at the
edge of the spectrum of the finite n Gaussian Unitary Ensemble (GUEn) and the
finite n expansion of Qn(x;t) and Pn(x;t). Using these large n expansions, we
give another proof of the derivation of an Edgeworth type theorem for the
largest eigenvalue distribution function of GUEn. We conclude with a brief
discussion on the derivation of the probability distribution function of the
corresponding largest eigenvalue in the Gaussian Orthogonal Ensemble (GOEn) and
Gaussian Symplectic Ensembles (GSEn)
Universal Parametric Correlations of Eigenvalues of Random Matrix Ensemble
Eigenvalue correlations of random matrix ensembles as a function of an
external perturbation are investigated vis the Dyson Brownian Motion Model in
the situation where the level density has a hard edge singularity. By solving a
linearized hydrodynamical equation, a universal dependence of the
density-density correlator on the external field is found. As an application we
obtain a formula for the variance of linear statistics with the parametric
dependence exhibited as a Laplace transform.Comment: 23 pages, late
Interpolation between Airy and Poisson statistics for unitary chiral non-Hermitian random matrix ensembles
We consider a family of chiral non-Hermitian Gaussian random matrices in the unitarily invariant symmetry class. The eigenvalue distribution in this model is expressed in terms of Laguerre polynomials in the complex plane. These are orthogonal with respect to a non-Gaussian weight including a modified Bessel function of the second kind, and we give an elementary proof for this. In the large n limit, the eigenvalue statistics at the spectral edge close to the real axis are described by the same family of kernels interpolating between Airy and Poisson that was recently found by one of the authors for the elliptic Ginibre ensemble. We conclude that this scaling limit is universal, appearing for two different non-Hermitian random matrix ensembles with unitary symmetry. As a second result we give an equivalent form for the interpolating Airy kernel in terms of a single real integral, similar to representations for the asymptotic kernel in the bulk and at the hard edge of the spectrum. This makes its structure as a one-parameter deformation of the Airy kernel more transparent
Ray helicity: a geometric invariant for multi-dimensional resonant wave conversion
For a multicomponent wave field propagating into a multidimensional
conversion region, the rays are shown to be helical, in general. For a
ray-based quantity to have a fundamental physical meaning it must be invariant
under two groups of transformations: congruence transformations (which shuffle
components of the multi-component wave field) and canonical transformations
(which act on the ray phase space). It is shown that for conversion between two
waves there is a new invariant not previously discussed: the intrinsic helicity
of the ray
Increasing subsequences and the hard-to-soft edge transition in matrix ensembles
Our interest is in the cumulative probabilities Pr(L(t) \le l) for the
maximum length of increasing subsequences in Poissonized ensembles of random
permutations, random fixed point free involutions and reversed random fixed
point free involutions. It is shown that these probabilities are equal to the
hard edge gap probability for matrix ensembles with unitary, orthogonal and
symplectic symmetry respectively. The gap probabilities can be written as a sum
over correlations for certain determinantal point processes. From these
expressions a proof can be given that the limiting form of Pr(L(t) \le l) in
the three cases is equal to the soft edge gap probability for matrix ensembles
with unitary, orthogonal and symplectic symmetry respectively, thereby
reclaiming theorems due to Baik-Deift-Johansson and Baik-Rains.Comment: LaTeX, 19 page
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