1,868 research outputs found
Topological expansion and boundary conditions
In this article, we compute the topological expansion of all possible
mixed-traces in a hermitian two matrix model. In other words we give a recipe
to compute the number of discrete surfaces of given genus, carrying an Ising
model, and with all possible given boundary conditions. The method is
recursive, and amounts to recursively cutting surfaces along interfaces. The
result is best represented in a diagrammatic way, and is thus rather simple to
use.Comment: latex, 25 pages. few misprints correcte
Loop equations for the semiclassical 2-matrix model with hard edges
The 2-matrix models can be defined in a setting more general than polynomial
potentials, namely, the semiclassical matrix model. In this case, the
potentials are such that their derivatives are rational functions, and the
integration paths for eigenvalues are arbitrary homology classes of paths for
which the integral is convergent. This choice includes in particular the case
where the integration path has fixed endpoints, called hard edges. The hard
edges induce boundary contributions in the loop equations. The purpose of this
article is to give the loop equations in that semicassical setting.Comment: Latex, 20 page
Mixed correlation functions in the 2-matrix model, and the Bethe ansatz
Using loop equation technics, we compute all mixed traces correlation
functions of the 2-matrix model to large N leading order. The solution turns
out to be a sort of Bethe Ansatz, i.e. all correlation functions can be
decomposed on products of 2-point functions. We also find that, when the
correlation functions are written collectively as a matrix, the loop equations
are equivalent to commutation relations.Comment: 38 pages, LaTex, 24 figures. misprints corrected, references added, a
technical part moved to appendi
Topological expansion of the 2-matrix model correlation functions: diagrammatic rules for a residue formula
We solve the loop equations of the hermitian 2-matrix model to all orders in
the topological expansion, i.e. we obtain all non-mixed correlation
functions, in terms of residues on an algebraic curve. We give two
representations of those residues as Feynman-like graphs, one of them involving
only cubic vertices.Comment: 48 pages, LaTex, 68 figure
Hamiltonian Cycles on a Random Three-coordinate Lattice
Consider a random three-coordinate lattice of spherical topology having 2v
vertices and being densely covered by a single closed, self-avoiding walk, i.e.
being equipped with a Hamiltonian cycle. We determine the number of such
objects as a function of v. Furthermore we express the partition function of
the corresponding statistical model as an elliptic integral.Comment: 10 pages, LaTeX, 3 eps-figures, one reference adde
Formal matrix integrals and combinatorics of maps
This article is a short review on the relationship between convergent matrix
integrals, formal matrix integrals, and combinatorics of maps. We briefly
summarize results developed over the last 30 years, as well as more recent
discoveries. We recall that formal matrix integrals are identical to
combinatorial generating functions for maps, and that formal matrix integrals
are in general very different from convergent matrix integrals. Finally, we
give a list of the classical matrix models which have played an important role
in physics in the past decades. Some of them are now well understood, some are
still difficult challenges.Comment: few misprints corrected, biblio modifie
Intersection numbers of spectral curves
We compute the symplectic invariants of an arbitrary spectral curve with only
1 branchpoint in terms of integrals of characteristic classes in the moduli
space of curves. Our formula associates to any spectral curve, a characteristic
class, which is determined by the laplace transform of the spectral curve. This
is a hint to the key role of Laplace transform in mirror symmetry. When the
spectral curve is y=\sqrt{x}, the formula gives Kontsevich--Witten intersection
numbers, when the spectral curve is chosen to be the Lambert function
\exp{x}=y\exp{-y}, the formula gives the ELSV formula for Hurwitz numbers, and
when one chooses the mirror of C^3 with framing f, i.e.
\exp{-x}=\exp{-yf}(1-\exp{-y}), the formula gives the Marino-Vafa formula, i.e.
the generating function of Gromov-Witten invariants of C^3. In some sense this
formula generalizes ELSV, Marino-Vafa formula, and Mumford formula.Comment: 53 pages, 1 fig, Latex, minor modification
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