325 research outputs found
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
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
Enumeration of maps with self avoiding loops and the O(n) model on random lattices of all topologies
We compute the generating functions of a O(n) model (loop gas model) on a
random lattice of any topology. On the disc and the cylinder, they were already
known, and here we compute all the other topologies. We find that the
generating functions (and the correlation functions of the lattice) obey the
topological recursion, as usual in matrix models, i.e they are given by the
symplectic invariants of their spectral curve.Comment: pdflatex, 89 pages, 12 labelled figures (15 figures at all), minor
correction
More on the exact solution of the O(n) model on a random lattice and an investigation of the case |n|>2
For the model on a random lattice has critical points to
which a scaling behaviour characteristic of 2D gravity interacting with
conformal matter fields with can be associated. Previously
we have written down an exact solution of this model valid at any point in the
coupling constant space and for any . The solution was parametrized in terms
of an auxiliary function. Here we determine the auxiliary function explicitly
as a combination of -functions, thereby completing the solution of the
model. Using our solution we investigate, for the simplest version of the
model, hitherto unexplored regions of the parameter space. For example we
determine in a closed form the eigenvalue density without any assumption of
being close to or at a critical point. This gives a generalization of the
Wigner semi-circle law to . We also study the model for . Both
for we find that the model is well defined in a certain region
of the coupling constant space. For we find no new critical points while
for we find new critical points at which the string susceptibility
exponent takes the value .Comment: 27 pages, LaTeX file (uses epsf) + 3 eps figures, formulas involving
the string susceptibility corrrected, no change in conclusion
An Iterative Solution of the Three-colour Problem on a Random Lattice
We study the generalisation of Baxter's three-colour problem to a random
lattice. Rephrasing the problem as a matrix model problem we discuss the
analyticity structure and the critical behaviour of the resulting matrix model.
Based on a set of loop equations we develop an algorithm which enables us to
solve the three-colour problem recursivelyComment: 14 pages, LaTeX, misprints corrected, approximation of (6.20) refine
Mixed Correlation Functions of the Two-Matrix Model
We compute the correlation functions mixing the powers of two non-commuting
random matrices within the same trace. The angular part of the integration was
partially known in the literature: we pursue the calculation and carry out the
eigenvalue integration reducing the problem to the construction of the
associated biorthogonal polynomials. The generating function of these
correlations becomes then a determinant involving the recursion coefficients of
the biorthogonal polynomials.Comment: 16 page
Hermitian Matrix Model with Plaquette Interaction
We study a hermitian -matrix model with plaquette interaction,
. By means of a conformal transformation we rewrite the
model as an model on a random lattice with a non polynomial potential.
This allows us to solve the model exactly. We investigate the critical
properties of the plaquette model and find that for the model
belongs to the same universality class as the model on a random lattice.Comment: 15 pages, no figures, two references adde
Correlation Functions of Complex Matrix Models
For a restricted class of potentials (harmonic+Gaussian potentials), we
express the resolvent integral for the correlation functions of simple traces
of powers of complex matrices of size , in term of a determinant; this
determinant is function of four kernels constructed from the orthogonal
polynomials corresponding to the potential and from their Cauchy transform. The
correlation functions are a sum of expressions attached to a set of fully
packed oriented loops configurations; for rotational invariant systems,
explicit expressions can be written for each configuration and more
specifically for the Gaussian potential, we obtain the large expansion ('t
Hooft expansion) and the so-called BMN limit.Comment: latex BMN.tex, 7 files, 6 figures, 30 pages (v2 for spelling mistake
and added reference) [http://www-spht.cea.fr/articles/T05/174
Mixed correlation function and spectral curve for the 2-matrix model
We compute the mixed correlation function in a way which involves only the
orthogonal polynomials with degrees close to , (in some sense like the
Christoffel Darboux theorem for non-mixed correlation functions). We also
derive new representations for the differential systems satisfied by the
biorthogonal polynomials, and we find new formulae for the spectral curve. In
particular we prove the conjecture of M. Bertola, claiming that the spectral
curve is the same curve which appears in the loop equations.Comment: latex, 1 figure, 55 page
Mirror of the refined topological vertex from a matrix model
We find an explicit matrix model computing the refined topological vertex,
starting from its representation in terms of plane partitions. We then find the
spectral curve of that matrix model, and thus the mirror symmetry of the
refined vertex. With the same method we also find a matrix model for the strip
geometry, and we find its mirror curve. The fact that there is a matrix model
shows that the refined topological string amplitudes also satisfy the
remodeling the B-model construction.Comment: pdflatex, 35 pages, 32 figure
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