3,309 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
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
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
Symmetrized models of last passage percolation and non-intersecting lattice paths
It has been shown that the last passage time in certain symmetrized models of
directed percolation can be written in terms of averages over random matrices
from the classical groups , and . We present a theory of
such results based on non-intersecting lattice paths, and integration
techniques familiar from the theory of random matrices. Detailed derivations of
probabilities relating to two further symmetrizations are also given.Comment: 21 pages, 5 figure
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
Andreev reflection from a topological superconductor with chiral symmetry
It was pointed out by Tewari and Sau that chiral symmetry (H -> -H if e
h) of the Hamiltonian of electron-hole (e-h) excitations in an N-mode
superconducting wire is associated with a topological quantum number
Q\in\mathbb{Z} (symmetry class BDI). Here we show that Q=Tr(r_{he}) equals the
trace of the matrix of Andreev reflection amplitudes, providing a link with the
electrical conductance G. We derive G=(2e^2/h)|Q| for |Q|=N,N-1, and more
generally provide a Q-dependent upper and lower bound on G. We calculate the
probability distribution P(G) for chaotic scattering, in the circular ensemble
of random-matrix theory, to obtain the Q-dependence of weak localization and
mesoscopic conductance fluctuations. We investigate the effects of chiral
symmetry breaking by spin-orbit coupling of the transverse momentum (causing a
class BDI-to-D crossover), in a model of a disordered semiconductor nanowire
with induced superconductivity. For wire widths less than the spin-orbit
coupling length, the conductance as a function of chemical potential can show a
sequence of 2e^2/h steps - insensitive to disorder.Comment: 10 pages, 5 figures. Corrected typo (missing square root) in
equations A13 and A1
Finite N Fluctuation Formulas for Random Matrices
For the Gaussian and Laguerre random matrix ensembles, the probability
density function (p.d.f.) for the linear statistic
is computed exactly and shown to satisfy a central limit theorem as . For the circular random matrix ensemble the p.d.f.'s for the linear
statistics and are calculated exactly by using a constant term identity
from the theory of the Selberg integral, and are also shown to satisfy a
central limit theorem as .Comment: LaTeX 2.09, 11 pages + 3 eps figs (needs epsf.sty
An adjoint for likelihood maximization
The process of likelihood maximization can be found in many different areas of computational modelling. However, the construction of such models via likelihood maximization requires the solution of a difficult multi-modal optimization problem involving an expensive O(n3) factorization. The optimization techniques used to solve this problem may require many such factorizations and can result in a significant bottle-neck. This article derives an adjoint formulation of the likelihood employed in the construction of a kriging model via reverse algorithmic differentiation. This adjoint is found to calculate the likelihood and all of its derivatives more efficiently than the standard analytical method and can therefore be utilised within a simple local search or within a hybrid global optimization to accelerate convergence and therefore reduce the cost of the likelihood optimization
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
Variance Calculations and the Bessel Kernel
In the Laguerre ensemble of N x N (positive) hermitian matrices, it is of
interest both theoretically and for applications to quantum transport problems
to compute the variance of a linear statistic, denoted var_N f, as N->infinity.
Furthermore, this statistic often contains an additional parameter alpha for
which the limit alpha->infinity is most interesting and most difficult to
compute numerically. We derive exact expressions for both lim_{N->infinity}
var_N f and lim_{alpha->infinity}lim_{N->infinity} var_N f.Comment: 7 pages; resubmitted to make postscript compatibl
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