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European Mathematical Society Publishing House
2024-03-19 02:37:29
3
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=AIHPD&vol=1&iss=2&update_since=2024-03-19
Annales de l’Institut Henri Poincaré D
Ann. Inst. Henri Poincaré Comb. Phys. Interact.
AIHPD
2308-5827
2308-5835
General
Combinatorics
Quantum theory
10.4171/AIHPD
http://www.ems-ph.org/doi/10.4171/AIHPD
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
1
2014
2
Planar maps, circle patterns and 2D gravity
François
David
CEA, Gif-sur-Yvette, FRANCE
Bertrand
Eynard
CEA Saclay, GIF-SUR-YVETTE CEDEX, FRANCE
Circle pattern, Random maps, Conformal invariance, Kähler geometry, 2D gravity, topological 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 Kähler metric over the space of Delaunay triangulations, whose prepotential has a simple formulation in term of ideal tessellations of the 3d hyperbolic space $\mathbb{H}_3$; (3) a discretized version (involving finite difference complex derivative operators $\nabla,\bar\nabla$) 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.
Combinatorics
Convex and discrete geometry
Probability theory and stochastic processes
Quantum theory
139
183
10.4171/AIHPD/5
http://www.ems-ph.org/doi/10.4171/AIHPD/5
An application of Khovanov homology to quantum codes
Benjamin
Audoux
Technopole Chateau Gombert, Marseille cedex 13, FRANCE
quantum codes, CSS codes, LDPC codes, Khovanov homology
We use Khovanov homology to define families of LDPC quantum error-correcting codes: unknot codes with asymptotical parameters $[[ \frac{3^{2 \ell+1}}{\sqrt{8\pi\ell}};1;2^\ell]]$; unlink codes with asymptotical parameters $[[\sqrt{\frac{3}{2\pi \ell}}6^\ell;2^\ell;2^\ell ]]$ and $(2,\ell)$-torus link codes with asymptotical parameters $[[n;1;d_n]]$ where $d_n>\frac{\sqrt{n}}{1.62}$.
Quantum theory
Manifolds and cell complexes
Information and communication, circuits
185
223
10.4171/AIHPD/6
http://www.ems-ph.org/doi/10.4171/AIHPD/6
Formal multidimensional integrals, stuffed maps, and topological recursion
Gaëtan
Borot
für Mathematik, BONN, GERMANY
Map enumeration, matrix models, 2D quantum gravity, loop equations, Tutte equation, topological recursion
We show that the large $N$ expansion in the multi-trace 1 formal hermitian matrix model is governed by a topological recursion with initial conditions. In terms of a 1$d$ gas of eigenvalues, this model includes – on top of the squared Vandermonde – multilinear interactions of any order between the eigenvalues. In this problem, the initial data ($\omega_1^0, \omega_2^0$) of the topological recursion is characterized: for $\omega_1^0$, by a non-linear, non-local Riemann-Hilbert problem on the discontinuity locus $\Gamma$ to determine; for $\omega_2^0$, by a related but linear, non-local Riemann-Hilbert problem on the discontinuity locus $\Gamma$. In combinatorics, this model enumerates discrete surfaces (maps) whose elementary 2-cells can have any topology - $\omega_1^0$ being the generating series of disks, $\omega_2^0$ that of cylinders. In particular, by substitution one may consider maps whose elementary cells are themselves maps, for which we propose the name ”stuffed maps”. In a sense, our results complete the program of the ”moment method” initiated in the 90s to compute the formal 1/$N$ in the one hermitian matrix model.
Combinatorics
Linear and multilinear algebra; matrix theory
Functions of a complex variable
225
264
10.4171/AIHPD/7
http://www.ems-ph.org/doi/10.4171/AIHPD/7