479 research outputs found
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
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
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
2-matrix versus complex matrix model, integrals over the unitary group as triangular integrals
We prove that the 2-hermitean matrix model and the complex-matrix model obey
the same loop equations, and as a byproduct, we find a formula for
Itzykzon-Zuber's type integrals over the unitary group. Integrals over U(n) are
rewritten as gaussian integrals over triangular matrices and then computed
explicitly. That formula is an efficient alternative to the former
Shatashvili's formula.Comment: 29 pages, Late
Topological expansion of beta-ensemble model and quantum algebraic geometry in the sectorwise approach
We solve the loop equations of the -ensemble model analogously to the
solution found for the Hermitian matrices . For \beta=1y^2=U(x)\beta((\hbar\partial)^2-U(x))\psi(x)=0\hbar\propto
(\sqrt\beta-1/\sqrt\beta)/Ny^2-U(x)[y,x]=\hbarF_h-expansion at arbitrary . The set of "flat"
coordinates comprises the potential times and the occupation numbers
\widetilde{\epsilon}_\alpha\mathcal F_0\widetilde{\epsilon}_\alpha$.Comment: 58 pages, 7 figure
Holomorphic anomaly and matrix models
The genus g free energies of matrix models can be promoted to modular
invariant, non-holomorphic amplitudes which only depend on the geometry of the
classical spectral curve. We show that these non-holomorphic amplitudes satisfy
the holomorphic anomaly equations of Bershadsky, Cecotti, Ooguri and Vafa. We
derive as well holomorphic anomaly equations for the open string sector. These
results provide evidence at all genera for the Dijkgraaf--Vafa conjecture
relating matrix models to type B topological strings on certain local
Calabi--Yau threefolds.Comment: 23 pages, LaTex, 3 figure
Large deviations of the maximal eigenvalue of random matrices
We present detailed computations of the 'at least finite' terms (three
dominant orders) of the free energy in a one-cut matrix model with a hard edge
a, in beta-ensembles, with any polynomial potential. beta is a positive number,
so not restricted to the standard values beta = 1 (hermitian matrices), beta =
1/2 (symmetric matrices), beta = 2 (quaternionic self-dual matrices). This
model allows to study the statistic of the maximum eigenvalue of random
matrices. We compute the large deviation function to the left of the expected
maximum. We specialize our results to the gaussian beta-ensembles and check
them numerically. Our method is based on general results and procedures already
developed in the literature to solve the Pastur equations (also called "loop
equations"). It allows to compute the left tail of the analog of Tracy-Widom
laws for any beta, including the constant term.Comment: 62 pages, 4 figures, pdflatex ; v2 bibliography corrected ; v3 typos
corrected and preprint added ; v4 few more numbers adde
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
A matrix model for the topological string II: The spectral curve and mirror geometry
In a previous paper, we presented a matrix model reproducing the topological
string partition function on an arbitrary given toric Calabi-Yau manifold.
Here, we study the spectral curve of our matrix model and thus derive, upon
imposing certain minimality assumptions on the spectral curve, the large volume
limit of the BKMP "remodeling the B-model" conjecture, the claim that
Gromov-Witten invariants of any toric Calabi-Yau 3-fold coincide with the
spectral invariants of its mirror curve.Comment: 1+37 page
Some properties of angular integrals
We find new representations for Itzykson-Zuber like angular integrals for
arbitrary beta, in particular for the orthogonal group O(n), the unitary group
U(n) and the symplectic group Sp(2n). We rewrite the Haar measure integral, as
a flat Lebesge measure integral, and we deduce some recursion formula on n. The
same methods gives also the Shatashvili's type moments. Finally we prove that,
in agreement with Brezin and Hikami's observation, the angular integrals are
linear combinations of exponentials whose coefficients are polynomials in the
reduced variables (x_i-x_j)(y_i-y_j).Comment: 43 pages, Late
- …