884 research outputs found
On the asymptotic behavior of a log gas in the bulk scaling limit in the presence of a varying external potential I
We study the determinant , of the integrable
Fredholm operator acting on the interval with kernel
. This
determinant arises in the analysis of a log-gas of interacting particles in the
bulk-scaling limit, at inverse temperature , in the presence of an
external potential supported on an interval of
length . We evaluate, in particular, the double scaling limit
of as and , in the
region , for
any fixed . This problem was first considered by Dyson in
\cite{Dy1}.Comment: 49 pages, 15 figures. Version 2 contains an extended introduction and
corrects typo
Large N duality beyond the genus expansion
We study non-perturbative aspects of the large N duality between Chern-Simons
theory and topological strings, and we find a rich structure of large N phase
transitions in the complex plane of the 't Hooft parameter. These transitions
are due to large N instanton effects, and they can be regarded as a deformation
of the Stokes phenomenon. Moreover, we show that, for generic values of the 't
Hooft coupling, instanton effects are not exponentially suppressed at large N
and they correct the genus expansion. This phenomenon was first discovered in
the context of matrix models, and we interpret it as a generalization of the
oscillatory asymptotics along anti-Stokes lines. In the string dual, the
instanton effects can be interpreted as corrections to the saddle string
geometry due to discretized neighboring geometries. As a mathematical
application, we obtain the 1/N asymptotics of the partition function of
Chern-Simons theory on L(2,1), and we test it numerically to high precision in
order to exhibit the importance of instanton effects.Comment: 37 pages, 24 figures. v2: clarifications and references added,
misprints corrected, to appear in JHE
Direct Integration and Non-Perturbative Effects in Matrix Models
We show how direct integration can be used to solve the closed amplitudes of
multi-cut matrix models with polynomial potentials. In the case of the cubic
matrix model, we give explicit expressions for the ring of non-holomorphic
modular objects that are needed to express all closed matrix model amplitudes.
This allows us to integrate the holomorphic anomaly equation up to holomorphic
modular terms that we fix by the gap condition up to genus four. There is an
one-dimensional submanifold of the moduli space in which the spectral curve
becomes the Seiberg--Witten curve and the ring reduces to the non-holomorphic
modular ring of the group . On that submanifold, the gap conditions
completely fix the holomorphic ambiguity and the model can be solved explicitly
to very high genus. We use these results to make precision tests of the
connection between the large order behavior of the 1/N expansion and
non-perturbative effects due to instantons. Finally, we argue that a full
understanding of the large genus asymptotics in the multi-cut case requires a
new class of non-perturbative sectors in the matrix model.Comment: 51 pages, 8 figure
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