7,554 research outputs found
Painleve II in random matrix theory and related fields
We review some occurrences of Painlev\'e II transcendents in the study of
two-dimensional Yang-Mills theory, fluctuation formulas for growth models, and
as distribution functions within random matrix theory. We first discuss
settings in which the parameter in the Painlev\'e equation is zero,
and the boundary condition is that of the Hasting-MacLeod solution. As well as
expressions involving the Painlev\'e transcendent itself, one encounters the
sigma form of the Painlev\'e II equation, and Lax pair equations in which the
Painlev\'e transcendent occurs as coefficients. We then consider settings which
give rise to general Painlev\'e II transcendents. In a particular
random matrix setting, new results for the corresponding boundary conditions in
the cases , 1 and 2 are presented.Comment: 25 pages, prepared for the special issue on Painleve equations in the
journal Constructive Approximatio
Discrete Painlev\'e equations and random matrix averages
The -function theory of Painlev\'e systems is used to derive
recurrences in the rank of certain random matrix averages over U(n). These
recurrences involve auxilary quantities which satisfy discrete Painlev\'e
equations. The random matrix averages include cases which can be interpreted as
eigenvalue distributions at the hard edge and in the bulk of matrix ensembles
with unitary symmetry. The recurrences are illustrated by computing the value
of a sequence of these distributions as varies, and demonstrating
convergence to the value of the appropriate limiting distribution.Comment: 25 page
Application of the -function theory of Painlev\'e equations to random matrices: \PVI, the JUE, CyUE, cJUE and scaled limits
Okamoto has obtained a sequence of -functions for the \PVI system
expressed as a double Wronskian determinant based on a solution of the Gauss
hypergeometric equation. Starting with integral solutions of the Gauss
hypergeometric equation, we show that the determinant can be re-expressed as
multi-dimensional integrals, and these in turn can be identified with averages
over the eigenvalue probability density function for the Jacobi unitary
ensemble (JUE), and the Cauchy unitary ensemble (CyUE) (the latter being
equivalent to the circular Jacobi unitary ensemble (cJUE)). Hence these
averages, which depend on four continuous parameters and the discrete parameter
, can be characterised as the solution of the second order second degree
equation satisfied by the Hamiltonian in the \PVI theory. Applications are
given to the evaluation of the spacing distribution for the circular unitary
ensemble (CUE) and its scaled counterpart, giving formulas more succinct than
those known previously; to expressions for the hard edge gap probability in the
scaled Laguerre orthogonal ensemble (LOE) (parameter a non-negative
integer) and Laguerre symplectic ensemble (LSE) (parameter an even
non-negative integer) as finite dimensional combinatorial integrals over the
symplectic and orthogonal groups respectively; to the evaluation of the
cumulative distribution function for the last passage time in certain models of
directed percolation; to the -function evaluation of the largest
eigenvalue in the finite LOE and LSE with parameter ; and to the
characterisation of the diagonal-diagonal spin-spin correlation in the
two-dimensional Ising model.Comment: AMSLate
Moments of the Gaussian Ensembles and the large- expansion of the densities
The loop equation formalism is used to compute the expansion of the
resolvent for the Gaussian ensemble up to and including the term at
. This allows the moments of the eigenvalue density to be computed
up to and including the 12-th power and the smoothed density to be expanded up
to and including the term at . The latter contain non-integrable
singularities at the endpoints of the support --- we show how to nonetheless
make sense of the average of a sufficiently smooth linear statistic. At the
special couplings , and there are characterisations of both
the resolvent and the moments which allows for the corresponding expansions to
be extended, in some recursive form at least, to arbitrary order. In this
regard we give fifth order linear differential equations for the density and
resolvent at and , which complements the known third order
linear differential equations for these quantities at .Comment: 30 pages, final version has been trimmed and additional moments have
been computed, which are recorded in the Appendi
On the Variance of the Index for the Gaussian Unitary Ensemble
We derive simple linear, inhomogeneous recurrences for the variance of the
index by utilising the fact that the generating function for the distribution
of the number of positive eigenvalues of a Gaussian unitary ensemble is a
-function of the fourth Painlev\'e equation. From this we deduce a simple
summation formula, several integral representations and finally an exact
hypergeometric function evaluation for the variance.Comment: Added references and authors dat
Discrete Painlev\'e equations, Orthogonal Polynomials on the Unit Circle and N-recurrences for averages over U(N) -- \PIIIa and \PV -functions
In this work we show that the Toeplitz determinants with the
symbols and -- known -functions for the \PIIIa and
\PV systems -- are characterised by nonlinear recurrences for the reflection
coefficients of the corresponding orthogonal polynomial system on the unit
circle. It is shown that these recurrences are entirely equivalent to the
discrete Painlev\'e equations associated with the degenerations of the rational
surfaces (discrete Painlev\'e {\rm II}) and (discrete Painlev\'e {\rm IV}) respectively
through the algebraic methodology based upon of the affine Weyl group symmetry
of the Painlev\'e system, originally due to Okamoto. In addition it is shown
that the difference equations derived by methods based upon the Toeplitz
lattice and Virasoro constraints, when reduced in order by exact summation, are
equivalent to our recurrences. Expressions in terms of generalised
hypergeometric functions are given for the reflection coefficients
respectively.Comment: AMS-Latex2e with AMS macro
Application of the -function theory of Painlev\'e equations to random matrices: \PV, \PIII, the LUE, JUE and CUE
With denoting an average with respect to the eigenvalue PDF for the
Laguerre unitary ensemble, the object of our study is
for and , where for and otherwise. Using Okamoto's development of the
theory of the Painlev\'e V equation, it is shown that is
a -function associated with the Hamiltonian therein, and so can be
characterised as the solution of a certain second order second degree
differential equation, or in terms of the solution of certain difference
equations. The cases and are of particular interest as they
correspond to the cumulative distribution and density function respectively for
the smallest and largest eigenvalue. In the case ,
is simply related to an average in the Jacobi unitary
ensemble, and this in turn is simply related to certain averages over the
orthogonal group, the unitary symplectic group and the circular unitary
ensemble. The latter integrals are of interest for their combinatorial content.
Also considered are the hard edge and soft edge scaled limits of
. In particular, in the hard edge scaled limit it is
shown that the limiting quantity can be evaluated
as a -function associated with the Hamiltonian in Okamoto's theory of the
Painlev\'e III equation.Comment: AMSLatex with CIMS macro
Discrete Painlev\'e equations for a class of \PVI -functions given as U(N) averages
In a recent work difference equations (Laguerre-Freud equations) for the
bi-orthogonal polynomials and related quantities corresponding to the weight on
the unit circle were derived.Here it
is shown that in the case these difference equations, when applied to
the calculation of the underlying U(N) average, reduce to a coupled system
identifiable with that obtained by Adler and van Moerbeke using methods of the
Toeplitz lattice and Virasoro constraints. Moreover it is shown that this
coupled system can be reduced to yield the discrete fifth Painlev\'e equation
\dPV as it occurs in the theory of the sixth Painlev\'e system. Methods based
on affine Weyl group symmetries of B\"acklund transformations have previously
yielded the \dPV equation but with different parameters for the same problem.
We find the explicit mapping between the two forms. Applications of our results
are made to give recurrences for the gap probabilities and moments in the
circular unitary ensemble of random matrices, and to the diagonal spin-spin
correlation function of the square lattice Ising model.Comment: Companion work to CA/041239
Exact Wigner surmise type evaluation of the spacing distribution in the bulk of the scaled random matrix ensembles
Random matrix ensembles with orthogonal and unitary symmetry correspond to
the cases of real symmetric and Hermitian random matrices respectively. We show
that the probability density function for the corresponding spacings between
consecutive eigenvalues can be written exactly in the Wigner surmise type form
for simply related to a Painlev\'e transcendent and
its anti-derivative. A formula consisting of the sum of two such terms is given
for the symplectic case (Hermitian matrices with real quaternion elements).Comment: 6 pages, Latex2
Singular Values of Products of Ginibre Random Matrices
The squared singular values of the product of complex Ginibre matrices
form a biorthogonal ensemble, and thus their distribution is fully determined
by a correlation kernel. The kernel permits a hard edge scaling to a form
specified in terms of certain Meijer G-functions, or equivalently
hypergeometric functions , also referred to as hyper-Bessel
functions. In the case it is well known that the corresponding gap
probability for no squared singular values in can be evaluated in terms
of a solution of a particular sigma form of the Painlev\'e III' system. One
approach to this result is a formalism due to Tracy and Widom, involving the
reduction of a certain integrable system. Strahov has generalised this
formalism to general , but has not exhibited its reduction. After
detailing the necessary working in the case , we consider the problem of
reducing the 12 coupled differential equations in the case to a single
differential equation for the resolvent. An explicit 4-th order nonlinear is
found for general hard edge parameters. For a particular choice of parameters,
evidence is given that this simplifies to a much simpler third order nonlinear
equation. The small and large asymptotics of the 4-th order equation are
discussed, as is a possible relationship of the systems to so-called
4-dimensional Painlev\'e-type equations.Comment: 33 pages, 4 figures, 1 tabl
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