5,656 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
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
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
Analytic solutions of the 1D finite coupling delta function Bose gas
An intensive study for both the weak coupling and strong coupling limits of
the ground state properties of this classic system is presented. Detailed
results for specific values of finite are given and from them results for
general are determined. We focus on the density matrix and concomitantly
its Fourier transform, the occupation numbers, along with the pair correlation
function and concomitantly its Fourier transform, the structure factor. These
are the signature quantities of the Bose gas. One specific result is that for
weak coupling a rational polynomial structure holds despite the transcendental
nature of the Bethe equations. All these new results are predicated on the
Bethe ansatz and are built upon the seminal works of the past.Comment: 23 pages, 0 figures, uses rotate.sty. A few lines added. Accepted by
Phys. Rev.
Random walks and random fixed-point free involutions
A bijection is given between fixed point free involutions of
with maximum decreasing subsequence size and two classes of vicious
(non-intersecting) random walker configurations confined to the half line
lattice points . In one class of walker configurations the maximum
displacement of the right most walker is . Because the scaled distribution
of the maximum decreasing subsequence size is known to be in the soft edge GOE
(random real symmetric matrices) universality class, the same holds true for
the scaled distribution of the maximum displacement of the right most walker.Comment: 10 page
- …