9,893 research outputs found
Non-homogenous disks in the chain of matrices
We investigate the generating functions of multi-colored discrete disks with
non-homogenous boundary conditions in the context of the Hermitian multi-matrix
model where the matrices are coupled in an open chain. We show that the study
of the spectral curve of the matrix model allows one to solve a set of loop
equations to get a recursive formula computing mixed trace correlation
functions to leading order in the large matrix limit.Comment: 25 pages, 4 figure
On Fenchel-Nielsen coordinates on Teichm\"uller spaces of surfaces of infinite type
We introduce Fenchel-Nielsen coordinates on Teicm\"uller spaces of surfaces
of infinite type. The definition is relative to a given pair of pants
decomposition of the surface. We start by establishing conditions under which
any pair of pants decomposition on a hyperbolic surface of infinite type can be
turned into a geometric decomposition, that is, a decomposition into hyperbolic
pairs of pants. This is expressed in terms of a condition we introduce and
which we call Nielsen convexity. This condition is related to Nielsen cores of
Fuchsian groups. We use this to define the Fenchel-Nielsen Teichm\"uller space
associated to a geometric pair of pants decomposition. We study a metric on
such a Teichm\"uller space, and we compare it to the quasiconformal
Teichm\"uller space, equipped with the Teichm\"uller metric. We study
conditions under which there is an equality between these Teichm\"uller spaces
and we study topological and metric properties of the identity map when this
map exists
Topology of quasiperiodic functions on the plane
The article describes a topological theory of quasiperiodic functions on the
plane. The development of this theory was started (in different terminology) by
the Moscow topology group in early 1980s. It was motivated by the needs of
solid state physics, as a partial (nongeneric) case of Hamiltonian foliations
of Fermi surfaces with multivalued Hamiltonian function. The unexpected
discoveries of their topological properties that were made in 1980s and 1990s
have finally led to nontrivial physical conclusions along the lines of the
so-called geometric strong magnetic field limit. A very fruitful new point of
view comes from the reformulation of that problem in terms of quasiperiodic
functions and an extension to higher dimensions made in 1999. One may say that,
for single crystal normal metals put in a magnetic field, the semiclassical
trajectories of electrons in the space of quasimomenta are exactly the level
lines of the quasiperiodic function with three quasiperiods that is the
dispersion relation restricted to a plane orthogonal to the magnetic field.
General studies of the topological properties of levels of quasiperiodic
functions on the plane with any number of quasiperiods were started in 1999
when certain ideas were formulated for the case of four quasiperiods. The last
section of this work contains a complete proof of these results. Some new
physical applications of the general problem were found recently.Comment: latex2e, 27 pages, 7 figure
Planar maps, circle patterns and 2d gravity
Via circle pattern techniques, random planar triangulations (with angle
variables) are mapped onto Delaunay triangulations in the complex plane. The
uniform measure on triangulations is mapped onto a conformally invariant
spatial point process. We show that this measure can be expressed as: (1) a sum
over 3-spanning-trees partitions of the edges of the Delaunay triangulations;
(2) the volume form of a K\"ahler metric over the space of Delaunay
triangulations, whose prepotential has a simple formulation in term of ideal
tessellations of the 3d hyperbolic space; (3) a discretized version (involving
finite difference complex derivative operators) of Polyakov's conformal
Fadeev-Popov determinant in 2d gravity; (4) a combination of Chern classes,
thus also establishing a link with topological 2d gravity.Comment: Misprints corrected and a couple of footnotes added. 42 pages, 17
figure
Rigid string instantons are pseudo-holomorphic curves
We show how to find explicit expressions for rigid string instantons for
general 4-manifold . It appears that they are pseudo-holomorphic curves in
the twistor space of . We present explicit formulae for . We
discuss their properties and speculate on relations to topology of 4-manifolds
and the theory of Yang-Mills fields.Comment: 18 pages,Late
Algebraic methods in random matrices and enumerative geometry
We review the method of symplectic invariants recently introduced to solve
matrix models loop equations, and further extended beyond the context of matrix
models. For any given spectral curve, one defined a sequence of differential
forms, and a sequence of complex numbers Fg . We recall the definition of the
invariants Fg, and we explain their main properties, in particular symplectic
invariance, integrability, modularity,... Then, we give several example of
applications, in particular matrix models, enumeration of discrete surfaces
(maps), algebraic geometry and topological strings, non-intersecting brownian
motions,...Comment: review article, Latex, 139 pages, many figure
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