3,893 research outputs found
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
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
Particles in a magnetic field and plasma analogies: doubly periodic boundary conditions
The -particle free fermion state for quantum particles in the plane
subject to a perpendicular magnetic field, and with doubly periodic boundary
conditions, is written in a product form. The absolute value of this is used to
formulate an exactly solvable one-component plasma model, and further motivates
the formulation of an exactly solvable two-species Coulomb gas. The large
expansion of the free energy of both these models exhibits the same O(1) term.
On the basis of a relationship to the Gaussian free field, this term is
predicted to be universal for conductive Coulomb systems in doubly periodic
boundary conditions.Comment: 12 page
Applications and generalizations of Fisher-Hartwig asymptotics
Fisher-Hartwig asymptotics refers to the large form of a class of
Toeplitz determinants with singular generating functions. This class of
Toeplitz determinants occurs in the study of the spin-spin correlations for the
two-dimensional Ising model, and the ground state density matrix of the
impenetrable Bose gas, amongst other problems in mathematical physics. We give
a new application of the original Fisher-Hartwig formula to the asymptotic
decay of the Ising correlations above , while the study of the Bose gas
density matrix leads us to generalize the Fisher-Hartwig formula to the
asymptotic form of random matrix averages over the classical groups and the
Gaussian and Laguerre unitary matrix ensembles. Another viewpoint of our
generalizations is that they extend to Hankel determinants the Fisher-Hartwig
asymptotic form known for Toeplitz determinants.Comment: 25 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
Correlations in two-component log-gas systems
A systematic study of the properties of particle and charge correlation
functions in the two-dimensional Coulomb gas confined to a one-dimensional
domain is undertaken. Two versions of this system are considered: one in which
the positive and negative charges are constrained to alternate in sign along
the line, and the other where there is no charge ordering constraint. Both
systems undergo a zero-density Kosterlitz-Thouless type transition as the
dimensionless coupling is varied through . In
the charge ordered system we use a perturbation technique to establish an
decay of the two-body correlations in the high temperature limit.
For , the low-fugacity expansion of the asymptotic
charge-charge correlation can be resummed to all orders in the fugacity. The
resummation leads to the Kosterlitz renormalization equations.Comment: 39 pages, 5 figures not included, Latex, to appear J. Stat. Phys.
Shortened version of abstract belo
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
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.
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