4,160 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
The Emergence of Superconducting Systems in Anti-de Sitter Space
In this article, we investigate the mathematical relationship between a (3+1)
dimensional gravity model inside Anti-de Sitter space , and a (2+1)
dimensional superconducting system on the asymptotically flat boundary of (in the absence of gravity). We consider a simple case of the Type II
superconducting model (in terms of Ginzburg-Landau theory) with an external
perpendicular magnetic field . An interaction potential is introduced
within the Lagrangian system. This provides more flexibility within the model,
when the superconducting system is close to the transition temperature .
Overall, our result demonstrates that the two Ginzburg-Landau differential
equations can be directly deduced from Einstein's theory of general relativity.Comment: 10 pages, 2 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
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
Modelling individual variability in cognitive development
Investigating variability in reasoning tasks can provide insights into key issues in the study of cognitive development. These include the mechanisms that underlie developmental transitions, and the distinction between individual differences and developmental disorders. We explored the mechanistic basis of variability in two connectionist models of cognitive development, a model of the Piagetian balance scale task (McClelland, 1989) and a model of the Piagetian conservation task (Shultz, 1998). For the balance scale task, we began with a simple feed-forward connectionist model and training patterns based on McClelland (1989). We investigated computational parameters, problem encodings, and training environments that contributed to variability in development, both across groups and within individuals. We report on the parameters that affect the complexity of reasoning and the nature of ‘rule’ transitions exhibited by networks learning to reason about balance scale problems. For the conservation task, we took the task structure and problem encoding of Shultz (1998) as our base model. We examined the computational parameters, problem encodings, and training environments that contributed to variability in development, in particular examining the parameters that affected the emergence of abstraction. We relate the findings to existing cognitive theories on the causes of individual differences in development
The lowest eigenvalue of Jacobi random matrix ensembles and Painlev\'e VI
We present two complementary methods, each applicable in a different range,
to evaluate the distribution of the lowest eigenvalue of random matrices in a
Jacobi ensemble. The first method solves an associated Painleve VI nonlinear
differential equation numerically, with suitable initial conditions that we
determine. The second method proceeds via constructing the power-series
expansion of the Painleve VI function. Our results are applied in a forthcoming
paper in which we model the distribution of the first zero above the central
point of elliptic curve L-function families of finite conductor and of
conjecturally orthogonal symmetry.Comment: 30 pages, 2 figure
The emergence of quantum capacitance in epitaxial graphene
We found an intrinsic redistribution of charge arises between epitaxial
graphene, which has intrinsically n-type doping, and an undoped substrate. In
particular, we studied in detail epitaxial graphene layers thermally elaborated
on C-terminated - (- ()). We have investigated
the charge distribution in graphene-substrate systems using Raman spectroscopy.
The influence of the substrate plasmons on the longitudinal optical phonons of
the substrates has been detected. The associated charge redistribution
reveals the formation of a capacitance between the graphene and the substrate.
Thus, we give for the first time direct evidence that the excess negative
charge in epitaxial monolayer graphene could be self-compensated by the
substrate without initial doping. This induced a previously unseen
redistribution of the charge-carrier density at the substrate-graphene
interface. There a quantum capacitor appears, without resorting to any
intentional external doping, as is fundamentally required for epitaxial
graphene. Although we have determined the electric field existing inside the
capacitor and revealed the presence of a minigap () for
epitaxial graphene on - face terminated carbon, it remains small in
comparison to that obtained for graphene on face terminated . The
fundamental electronic properties found here in graphene on substrates
may be important for developing the next generation of quantum technologies and
electronic/plasmonic devices.Comment: 26 pages, 8 figures, available online as uncorrected proof, Journal
of Materials Chemistry C (2016
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
The plasma picture of the fractional quantum Hall effect with internal SU(K) symmetries
We consider trial wavefunctions exhibiting SU(K) symmetry which may be
well-suited to grasp the physics of the fractional quantum Hall effect with
internal degrees of freedom. Systems of relevance may be either
spin-unpolarized states (K=2), semiconductors bilayers (K=2,4) or graphene
(K=4). We find that some introduced states are unstable, undergoing phase
separation or phase transition. This allows us to strongly reduce the set of
candidate wavefunctions eligible for a particular filling factor. The stability
criteria are obtained with the help of Laughlin's plasma analogy, which we
systematically generalize to the multicomponent SU(K) case. The validity of
these criteria are corroborated by exact-diagonalization studies, for SU(2) and
SU(4). Furthermore, we study the pair-correlation functions of the ground state
and elementary charged excitations within the multicomponent plasma picture.Comment: 13 pages, 7 figures; reference added, accepted for publication in PR
Scaling limit of vicious walks and two-matrix model
We consider the diffusion scaling limit of the one-dimensional vicious walker
model of Fisher and derive a system of nonintersecting Brownian motions. The
spatial distribution of particles is studied and it is described by use of
the probability density function of eigenvalues of Gaussian random
matrices. The particle distribution depends on the ratio of the observation
time and the time interval in which the nonintersecting condition is
imposed. As is going on from 0 to 1, there occurs a transition of
distribution, which is identified with the transition observed in the
two-matrix model of Pandey and Mehta. Despite of the absence of matrix
structure in the original vicious walker model, in the diffusion scaling limit,
accumulation of contact repulsive interactions realizes the correlated
distribution of eigenvalues in the multimatrix model as the particle
distribution.Comment: REVTeX4, 12 pages, no figure, minor corrections made for publicatio
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