6,560 research outputs found
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
Derivation of an eigenvalue probability density function relating to the Poincare disk
A result of Zyczkowski and Sommers [J.Phys.A, 33, 2045--2057 (2000)] gives
the eigenvalue probability density function for the top N x N sub-block of a
Haar distributed matrix from U(N+n). In the case n \ge N, we rederive this
result, starting from knowledge of the distribution of the sub-blocks,
introducing the Schur decomposition, and integrating over all variables except
the eigenvalues. The integration is done by identifying a recursive structure
which reduces the dimension. This approach is inspired by an analogous approach
which has been recently applied to determine the eigenvalue probability density
function for random matrices A^{-1} B, where A and B are random matrices with
entries standard complex normals. We relate the eigenvalue distribution of the
sub-blocks to a many body quantum state, and to the one-component plasma, on
the pseudosphere.Comment: 11 pages; To appear in J.Phys
Jacobians and rank 1 perturbations relating to unitary Hessenberg matrices
In a recent work Killip and Nenciu gave random recurrences for the
characteristic polynomials of certain unitary and real orthogonal upper
Hessenberg matrices. The corresponding eigenvalue p.d.f.'s are
beta-generalizations of the classical groups. Left open was the direct
calculation of certain Jacobians. We provide the sought direct calculation.
Furthermore, we show how a multiplicative rank 1 perturbation of the unitary
Hessenberg matrices provides a joint eigenvalue p.d.f generalizing the circular
beta-ensemble, and we show how this joint density is related to known
inter-relations between circular ensembles. Projecting the joint density onto
the real line leads to the derivation of a random three-term recurrence for
polynomials with zeros distributed according to the circular Jacobi
beta-ensemble.Comment: 23 page
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
A random matrix decimation procedure relating to
Classical random matrix ensembles with orthogonal symmetry have the property
that the joint distribution of every second eigenvalue is equal to that of a
classical random matrix ensemble with symplectic symmetry. These results are
shown to be the case of a family of inter-relations between eigenvalue
probability density functions for generalizations of the classical random
matrix ensembles referred to as -ensembles. The inter-relations give
that the joint distribution of every -st eigenvalue in certain
-ensembles with is equal to that of another
-ensemble with . The proof requires generalizing a
conditional probability density function due to Dixon and Anderson.Comment: 19 pages, 1 figur
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
The Ideal Conductor Limit
This paper compares two methods of statistical mechanics used to study a
classical Coulomb system S near an ideal conductor C. The first method consists
in neglecting the thermal fluctuations in the conductor C and constrains the
electric potential to be constant on it. In the second method the conductor C
is considered as a conducting Coulomb system the charge correlation length of
which goes to zero. It has been noticed in the past, in particular cases, that
the two methods yield the same results for the particle densities and
correlations in S. It is shown that this is true in general for the quantities
which depend only on the degrees of freedom of S, but that some other
quantities, especially the electric potential correlations and the stress
tensor, are different in the two approaches. In spite of this the two methods
give the same electric forces exerted on S.Comment: 19 pages, plain TeX. Submited to J. Phys. A: Math. Ge
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