345 research outputs found
Fisher Hartwig determinants, conformal field theory and universality in generalised XX models
We discuss certain quadratic models of spinless fermions on a 1D lattice, and
their corresponding spin chains. These were studied by Keating and Mezzadri in
the context of their relation to the Haar measures of the classical compact
groups. We show how these models correspond to translation invariant models on
an infinite or semi-infinite chain, which in the simplest case reduce to the
familiar XX model. We give physical context to mathematical results for the
entanglement entropy, and calculate the spin-spin correlation functions using
the Fisher-Hartwig conjecture. These calculations rigorously demonstrate
universality in classes of these models. We show that these are in agreement
with field theoretic and renormalization group arguments that we provide
Hankel determinant and orthogonal polynomials for a Gaussian weight with a discontinuity at the edge
We compute asymptotics for Hankel determinants and orthogonal polynomials
with respect to a discontinuous Gaussian weight, in a critical regime where the
discontinuity is close to the edge of the associated equilibrium measure
support. Their behavior is described in terms of the Ablowitz-Segur family of
solutions to the Painlev\'e II equation. Our results complement the ones in
[Xu,Zhao,2011]. As consequences of our results, we conjecture asymptotics for
an Airy kernel Fredholm determinant and total integral identities for
Painlev\'e II transcendents, and we also prove a new result on the poles of the
Ablowitz-Segur solutions to the Painlev\'e II equation. We also highlight
applications of our results in random matrix theory.Comment: 35 pages, 4 figure
Connection problem for the sine-Gordon/Painlev\'e III tau function and irregular conformal blocks
The short-distance expansion of the tau function of the radial
sine-Gordon/Painlev\'e III equation is given by a convergent series which
involves irregular conformal blocks and possesses certain periodicity
properties with respect to monodromy data. The long-distance irregular
expansion exhibits a similar periodicity with respect to a different pair of
coordinates on the monodromy manifold. This observation is used to conjecture
an exact expression for the connection constant providing relative
normalization of the two series. Up to an elementary prefactor, it is given by
the generating function of the canonical transformation between the two sets of
coordinates.Comment: 18 pages, 1 figur
Asymptotic Gap Probability Distributions of the Gaussian Unitary Ensembles and Jacobi Unitary Ensembles
In this paper, we address a class of problems in unitary ensembles.
Specifically, we study the probability that a gap symmetric about 0, i.e.
is found in the Gaussian unitary ensembles (GUE) and the Jacobi
unitary ensembles (JUE) (where in the JUE, we take the parameters
). By exploiting the even parity of the weight, a doubling of the
interval to for the GUE, and , for the (symmetric) JUE,
shows that the gap probabilities maybe determined as the product of the
smallest eigenvalue distributions of the LUE with parameter and
and the (shifted) JUE with weights and
The function, namely, the derivative of the
log of the smallest eigenvalue distributions of the finite- LUE or the JUE,
satisfies the Jimbo-Miwa-Okamoto form of and ,
although in the shift Jacobi case, with the weight
the parameter does not show up in the equation. We also obtain the
asymptotic expansions for the smallest eigenvalue distributions of the Laguerre
unitary and Jacobi unitary ensembles after appropriate double scalings, and
obtained the constants in the asymptotic expansion of the gap probablities,
expressed in term of the Barnes function valuated at special point.Comment: 38 page
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