7,580 research outputs found
The sine-law gap probability, Painlev\'e 5, and asymptotic expansion by the topological recursion
The goal of this article is to rederive the connection between the Painlev\'e
integrable system and the universal eigenvalues correlation functions of
double-scaled hermitian matrix models, through the topological recursion
method. More specifically we prove, \textbf{to all orders}, that the WKB
asymptotic expansions of the -function as well as of determinantal
formulas arising from the Painlev\'e Lax pair are identical to the large
double scaling asymptotic expansions of the partition function and
correlation functions of any hermitian matrix model around a regular point in
the bulk. In other words, we rederive the "sine-law" universal bulk asymptotic
of large random matrices and provide an alternative perturbative proof of
universality in the bulk with only algebraic methods. Eventually we exhibit the
first orders of the series expansion up to .Comment: 37 pages, 1 figure, published in Random Matrices: Theory and
Application
Solving the 3D Ising Model with the Conformal Bootstrap
We study the constraints of crossing symmetry and unitarity in general 3D
Conformal Field Theories. In doing so we derive new results for conformal
blocks appearing in four-point functions of scalars and present an efficient
method for their computation in arbitrary space-time dimension. Comparing the
resulting bounds on operator dimensions and OPE coefficients in 3D to known
results, we find that the 3D Ising model lies at a corner point on the boundary
of the allowed parameter space. We also derive general upper bounds on the
dimensions of higher spin operators, relevant in the context of theories with
weakly broken higher spin symmetries.Comment: 32 pages, 11 figures; v2: refs added, small changes in Section 5.3,
Fig. 7 replaced; v3: ref added, fits redone in Section 5.
Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials
We prove that every rational extension of the quantum harmonic oscillator
that is exactly solvable by polynomials is monodromy free, and therefore can be
obtained by applying a finite number of state-deleting Darboux transformations
on the harmonic oscillator. Equivalently, every exceptional orthogonal
polynomial system of Hermite type can be obtained by applying a Darboux-Crum
transformation to the classical Hermite polynomials. Exceptional Hermite
polynomial systems only exist for even codimension 2m, and they are indexed by
the partitions \lambda of m. We provide explicit expressions for their
corresponding orthogonality weights and differential operators and a separate
proof of their completeness. Exceptional Hermite polynomials satisfy a 2l+3
recurrence relation where l is the length of the partition \lambda. Explicit
expressions for such recurrence relations are given.Comment: 25 pages, typed in AMSTe
Tree expansion in time-dependent perturbation theory
The computational complexity of time-dependent perturbation theory is
well-known to be largely combinatorial whatever the chosen expansion method and
family of parameters (combinatorial sequences, Goldstone and other Feynman-type
diagrams...). We show that a very efficient perturbative expansion, both for
theoretical and numerical purposes, can be obtained through an original
parametrization by trees and generalized iterated integrals. We emphasize above
all the simplicity and naturality of the new approach that links perturbation
theory with classical and recent results in enumerative and algebraic
combinatorics. These tools are applied to the adiabatic approximation and the
effective Hamiltonian. We prove perturbatively and non-perturbatively the
convergence of Morita's generalization of the Gell-Mann and Low wavefunction.
We show that summing all the terms associated to the same tree leads to an
utter simplification where the sum is simpler than any of its terms. Finally,
we recover the time-independent equation for the wave operator and we give an
explicit non-recursive expression for the term corresponding to an arbitrary
tree.Comment: 22 pages, 2 figure
On the local systems Hamiltonian in the weakly nonlocal Poisson brackets
We study in this work the important class of nonlocal Poisson Brackets (PB)
which we call weakly nonlocal. They appeared recently in some investigations in
the Soliton Theory. However there was no theory of such brackets except very
special first order case. Even in this case the theory was not developed
enough. In particular, we introduce the Physical forms and find Casimirs,
Momentum and Canonical forms for the most important Hydrodynamic type PB of
that kind and their dependence on the boundary conditions.Comment: 45 pages, late
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