4,520 research outputs found
A Combinatorial Interpretation of the Free Fermion Condition of the Six-Vertex Model
The free fermion condition of the six-vertex model provides a 5 parameter
sub-manifold on which the Bethe Ansatz equations for the wavenumbers that enter
into the eigenfunctions of the transfer matrices of the model decouple, hence
allowing explicit solutions. Such conditions arose originally in early
field-theoretic S-matrix approaches. Here we provide a combinatorial
explanation for the condition in terms of a generalised Gessel-Viennot
involution. By doing so we extend the use of the Gessel-Viennot theorem,
originally devised for non-intersecting walks only, to a special weighted type
of \emph{intersecting} walk, and hence express the partition function of
such walks starting and finishing at fixed endpoints in terms of the single
walk partition functions
Growth models, random matrices and Painleve transcendents
The Hammersley process relates to the statistical properties of the maximum
length of all up/right paths connecting random points of a given density in the
unit square from (0,0) to (1,1). This process can also be interpreted in terms
of the height of the polynuclear growth model, or the length of the longest
increasing subsequence in a random permutation. The cumulative distribution of
the longest path length can be written in terms of an average over the unitary
group. Versions of the Hammersley process in which the points are constrained
to have certain symmetries of the square allow similar formulas. The derivation
of these formulas is reviewed. Generalizing the original model to have point
sources along two boundaries of the square, and appropriately scaling the
parameters gives a model in the KPZ universality class. Following works of Baik
and Rains, and Pr\"ahofer and Spohn, we review the calculation of the scaled
cumulative distribution, in which a particular Painlev\'e II transcendent plays
a prominent role.Comment: 27 pages, 5 figure
Random Matrix Theory and the Sixth Painlev\'e Equation
A feature of certain ensembles of random matrices is that the corresponding
measure is invariant under conjugation by unitary matrices. Study of such
ensembles realised by matrices with Gaussian entries leads to statistical
quantities related to the eigenspectrum, such as the distribution of the
largest eigenvalue, which can be expressed as multidimensional integrals or
equivalently as determinants. These distributions are well known to be
-functions for Painlev\'e systems, allowing for the former to be
characterised as the solution of certain nonlinear equations. We consider the
random matrix ensembles for which the nonlinear equation is the form
of \PVI. Known results are reviewed, as is their implication by way of series
expansions for the distributions. New results are given for the boundary
conditions in the neighbourhood of the fixed singularities at of
\PVI displayed by a generalisation of the generating function for the
distributions. The structure of these expansions is related to Jimbo's general
expansions for the -function of \PVI in the neighbourhood of its
fixed singularities, and this theory is itself put in its context of the linear
isomonodromy problem relating to \PVI.Comment: Dedicated to the centenary of the publication of the Painlev\'e VI
equation in the Comptes Rendus de l'Academie des Sciences de Paris by Richard
Fuchs in 190
{\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
Reunion of Vicious Walkers: Results from -Expansion -
The anomalous exponent, , for the decay of the reunion probability
of vicious walkers, each of length , in dimensions,
is shown to come from the multiplicative renormalization constant of a
directed polymer partition function. Using renormalization group(RG) we
evaluate to . The survival probability exponent is
. For , our RG is exact and stops at .
For , the log corrections are also determined. The number of walkers that
are sure to reunite is 2 and has no expansion.Comment: No of pages: 11, 1figure on request, Revtex3,IP/BBSR/929
Isomonodromic deformation theory and the next-to-diagonal correlations of the anisotropic square lattice Ising model
In 1980 Jimbo and Miwa evaluated the diagonal two-point correlation function
of the square lattice Ising model as a -function of the sixth Painlev\'e
system by constructing an associated isomonodromic system within their theory
of holonomic quantum fields. More recently an alternative isomonodromy theory
was constructed based on bi-orthogonal polynomials on the unit circle with
regular semi-classical weights, for which the diagonal Ising correlations arise
as the leading coefficient of the polynomials specialised appropriately. Here
we demonstrate that the next-to-diagonal correlations of the anisotropic Ising
model are evaluated as one of the elements of this isomonodromic system or
essentially as the Cauchy-Hilbert transform of one of the bi-orthogonal
polynomials.Comment: 11 pages, 1 figur
Boundary conditions associated with the Painlev\'e III' and V evaluations of some random matrix averages
In a previous work a random matrix average for the Laguerre unitary ensemble,
generalising the generating function for the probability that an interval at the hard edge contains eigenvalues, was evaluated in terms of
a Painlev\'e V transcendent in -form. However the boundary conditions
for the corresponding differential equation were not specified for the full
parameter space. Here this task is accomplished in general, and the obtained
functional form is compared against the most general small behaviour of
the Painlev\'e V equation in -form known from the work of Jimbo. An
analogous study is carried out for the the hard edge scaling limit of the
random matrix average, which we have previously evaluated in terms of a
Painlev\'e \IIId transcendent in -form. An application of the latter
result is given to the rapid evaluation of a Hankel determinant appearing in a
recent work of Conrey, Rubinstein and Snaith relating to the derivative of the
Riemann zeta function
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