4,324 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
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
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 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
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
Spectral density asymptotics for Gaussian and Laguerre -ensembles in the exponentially small region
The first two terms in the large asymptotic expansion of the
moment of the characteristic polynomial for the Gaussian and Laguerre
-ensembles are calculated. This is used to compute the asymptotic
expansion of the spectral density in these ensembles, in the exponentially
small region outside the leading support, up to terms . The leading form
of the right tail of the distribution of the largest eigenvalue is given by the
density in this regime. It is demonstrated that there is a scaling from this,
to the right tail asymptotics for the distribution of the largest eigenvalue at
the soft edge.Comment: 19 page
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Gut microbial features can predict host phenotype response to protein deficiency.
Malnutrition remains a major health problem in low- and middle-income countries. During low protein intake, <0.67 g/kg/day, there is a loss of nitrogen (N2 ) balance, due to the unavailability of amino acid for metabolism and unbalanced protein catabolism results. However, there are individuals, who consume the same low protein intake, and preserve N2 balance for unknown reasons. A novel factor, the gut microbiota, may account for these N2 balance differences. To investigate this, we correlated gut microbial profiles with the growth of four murine strains (C57Bl6/J, CD-1, FVB, and NIH-Swiss) on protein deficient (PD) diet. Results show that a PD diet exerts a strain-dependent impact on growth and N2 balance as determined through analysis of urinary urea, ammonia and creatinine excretion. Bacterial alpha diversity was significantly (P < 0.05, FDR) lower across all strains on a PD diet compared to normal chow (NC). Multi-group analyses of the composition of microbiomes (ANCOM) revealed significantly differential microbial signatures between the four strains independent of diet. However, mice on a PD diet demonstrated differential enrichment of bacterial genera including, Allobaculum (C57Bl6/J), Parabacteroides (CD-1), Turicibacter (FVB), and Mucispirillum (NIH-Swiss) relative to NC. For instance, selective comparison of the CD-1 (gained weight) and C57Bl6/J (did not gain weight) strains on PD diet also demonstrated significant pathway enrichment of dihydroorodate dehydrogenase, rRNA methyltransferases, and RNA splicing ligase in the CD-1 strains compared to C57Bl6/J strains; which might account in their ability to retain growth despite a protein deficient diet. Taken together, these results suggest a potential relationship between the specific gut microbiota, N2 balance and animal response to malnutrition
Upgrading School Buildings in Mexico with Social Participation: The Better Schools Programme
This review of Mexico’s Better Schools Programme was conducted in 2012 by the OECD Centre for Effective Learning Environments (CELE). In 2008, the federal government created the Programme to repair and improve the physical infrastructure of schools for basic education throughout Mexico. A key characteristic of the programme is social participation and the engagement of the each school community. The review team’s recommendations offer lessons to all governments investing in educational infrastructure to improve the quality of education
Eigenvalue distributions for some correlated complex sample covariance matrices
The distributions of the smallest and largest eigenvalues for the matrix
product , where is an complex Gaussian matrix
with correlations both along rows and down columns, are expressed as determinants. In the case of correlation along rows, these expressions are
computationally more efficient than those involving sums over partitions and
Schur polynomials reported recently for the same distributions.Comment: 11 page
Gap Probabilities for Edge Intervals in Finite Gaussian and Jacobi Unitary Matrix Ensembles
The probabilities for gaps in the eigenvalue spectrum of the finite dimension
random matrix Hermite and Jacobi unitary ensembles on some
single and disconnected double intervals are found. These are cases where a
reflection symmetry exists and the probability factors into two other related
probabilities, defined on single intervals. Our investigation uses the system
of partial differential equations arising from the Fredholm determinant
expression for the gap probability and the differential-recurrence equations
satisfied by Hermite and Jacobi orthogonal polynomials. In our study we find
second and third order nonlinear ordinary differential equations defining the
probabilities in the general case. For N=1 and N=2 the probabilities and
thus the solution of the equations are given explicitly. An asymptotic
expansion for large gap size is obtained from the equation in the Hermite case,
and also studied is the scaling at the edge of the Hermite spectrum as , and the Jacobi to Hermite limit; these last two studies make
correspondence to other cases reported here or known previously. Moreover, the
differential equation arising in the Hermite ensemble is solved in terms of an
explicit rational function of a {Painlev\'e-V} transcendent and its derivative,
and an analogous solution is provided in the two Jacobi cases but this time
involving a {Painlev\'e-VI} transcendent.Comment: 32 pages, Latex2
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