19 research outputs found
The Combinatorics of Alternating Tangles: from theory to computerized enumeration
We study the enumeration of alternating links and tangles, considered up to
topological (flype) equivalences. A weight is given to each connected
component, and in particular the limit yields information about
(alternating) knots. Using a finite renormalization scheme for an associated
matrix model, we first reduce the task to that of enumerating planar
tetravalent diagrams with two types of vertices (self-intersections and
tangencies), where now the subtle issue of topological equivalences has been
eliminated. The number of such diagrams with vertices scales as for
. We next show how to efficiently enumerate these diagrams (in time
) by using a transfer matrix method. We give results for various
generating functions up to 22 crossings. We then comment on their large-order
asymptotic behavior.Comment: proceedings European Summer School St-Petersburg 200
Eigenvalues of PT-symmetric oscillators with polynomial potentials
We study the eigenvalue problem
with the boundary
conditions that decays to zero as tends to infinity along the rays
, where is a polynomial and integers . We provide an
asymptotic expansion of the eigenvalues as , and prove
that for each {\it real} polynomial , the eigenvalues are all real and
positive, with only finitely many exceptions.Comment: 23 pages, 1 figure. v2: equation (14) as well as a few subsequent
equations has been changed. v3: typos correcte
Non linear pseudo-bosons versus hidden Hermiticity. II: The case of unbounded operators
Parallels between the notions of nonlinear pseudobosons and of an apparent
non-Hermiticity of observables as shown in paper I (arXiv: 1109.0605) are
demonstrated to survive the transition to the quantum models based on the use
of unbounded metric in the Hilbert space of states.Comment: 21 p
Correlation functions of eigenvalues of multi-matrix models, and the limit of a time dependent matrix
We consider the correlation functions of eigenvalues of a unidimensional
chain of large random hermitian matrices. An asymptotic expression of the
orthogonal polynomials allows to find new results for the correlations of
eigenvalues of different matrices of the chain. Eventually, we consider the
limit of the infinite chain of matrices, which can be interpreted as a time
dependent one-matrix model, and give the correlation functions of eigenvalues
at different times.Comment: Tex-Harvmac, 27 pages, submitted to Journ. Phys.
Breakdown of universality in multi-cut matrix models
We solve the puzzle of the disagreement between orthogonal polynomials
methods and mean field calculations for random NxN matrices with a disconnected
eigenvalue support. We show that the difference does not stem from a Z2
symmetry breaking, but from the discreteness of the number of eigenvalues. This
leads to additional terms (quasiperiodic in N) which must be added to the naive
mean field expressions. Our result invalidates the existence of a smooth
topological large N expansion and some postulated universality properties of
correlators. We derive the large N expansion of the free energy for the general
2-cut case. From it we rederive by a direct and easy mean-field-like method the
2-point correlators and the asymptotic orthogonal polynomials. We extend our
results to any number of cuts and to non-real potentials.Comment: 35 pages, Latex (1 file) + 3 figures (3 .eps files), revised to take
into account a few reference
An exactly solvable quantum-lattice model with a tunable degree of nonlocality
An array of N subsequent Laguerre polynomials is interpreted as an
eigenvector of a non-Hermitian tridiagonal Hamiltonian with real spectrum
or, better said, of an exactly solvable N-site-lattice cryptohermitian
Hamiltonian whose spectrum is known as equal to the set of zeros of the N-th
Laguerre polynomial. The two key problems (viz., the one of the ambiguity and
the one of the closed-form construction of all of the eligible inner products
which make Hermitian in the respective {\em ad hoc} Hilbert spaces) are
discussed. Then, for illustration, the first four simplest, parametric
definitions of inner products with and are explicitly
displayed. In mathematical terms these alternative inner products may be
perceived as alternative Hermitian conjugations of the initial N-plet of
Laguerre polynomials. In physical terms the parameter may be interpreted as
a measure of the "smearing of the lattice coordinates" in the model.Comment: 35 p
Non linear pseudo-bosons versus hidden Hermiticity
The increasingly popular concept of a hidden Hermiticity of operators (i.e.,
of their Hermiticity with respect to an {\it ad hoc} inner product in Hilbert
space) is compared with the recently introduced notion of {\em non-linear
pseudo-bosons}. The formal equivalence between these two notions is deduced
under very general assumptions. Examples of their applicability in quantum
mechanics are discussed.Comment: 20 p
Fermionic Quantum Gravity
We study the statistical mechanics of random surfaces generated by NxN
one-matrix integrals over anti-commuting variables. These Grassmann-valued
matrix models are shown to be equivalent to NxN unitary versions of generalized
Penner matrix models. We explicitly solve for the combinatorics of 't Hooft
diagrams of the matrix integral and develop an orthogonal polynomial
formulation of the statistical theory. An examination of the large N and double
scaling limits of the theory shows that the genus expansion is a Borel summable
alternating series which otherwise coincides with two-dimensional quantum
gravity in the continuum limit. We demonstrate that the partition functions of
these matrix models belong to the relativistic Toda chain integrable hierarchy.
The corresponding string equations and Virasoro constraints are derived and
used to analyse the generalized KdV flow structure of the continuum limit.Comment: 59 pages LaTeX, 1 eps figure. Uses epsf. References and
acknowledgments adde