Combinatorial Identities from the Spectral Theory of Quantum Graphs

We present a few combinatorial identities which were encountered in our work on the spectral theory of quantum graphs. They establish a new connection between the theory of random matrix ensembles and combinatorics.Comment: 16 pages, RevTeX, 1 figur

Combinatorial identities for binary necklaces from exact ray-splitting trace formulae

Based on an exact trace formula for a one-dimensional ray-splitting system, we derive novel combinatorial identities for cyclic binary sequences (P\'olya necklaces).Comment: 15 page

Trace Formulae and Spectral Statistics for Discrete Laplacians on Regular Graphs (I)

Trace formulae for d-regular graphs are derived and used to express the spectral density in terms of the periodic walks on the graphs under consideration. The trace formulae depend on a parameter w which can be tuned continuously to assign different weights to different periodic orbit contributions. At the special value w=1, the only periodic orbits which contribute are the non back- scattering orbits, and the smooth part in the trace formula coincides with the Kesten-McKay expression. As w deviates from unity, non vanishing weights are assigned to the periodic walks with back-scatter, and the smooth part is modified in a consistent way. The trace formulae presented here are the tools to be used in the second paper in this sequence, for showing the connection between the spectral properties of d-regular graphs and the theory of random matrices.Comment: 22 pages, 3 figure

Trace identities and their semiclassical implications

The compatibility of the semiclassical quantization of area-preserving maps with some exact identities which follow from the unitarity of the quantum evolution operator is discussed. The quantum identities involve relations between traces of powers of the evolution operator. For classically {\it integrable} maps, the semiclassical approximation is shown to be compatible with the trace identities. This is done by the identification of stationary phase manifolds which give the main contributions to the result. The same technique is not applicable for {\it chaotic} maps, and the compatibility of the semiclassical theory in this case remains unsettled. The compatibility of the semiclassical quantization with the trace identities demonstrates the crucial importance of non-diagonal contributions.Comment: LaTeX - IOP styl

Conformal Field Theories, Graphs and Quantum Algebras

This article reviews some recent progress in our understanding of the structure of Rational Conformal Field Theories, based on ideas that originate for a large part in the work of A. Ocneanu. The consistency conditions that generalize modular invariance for a given RCFT in the presence of various types of boundary conditions --open, twisted-- are encoded in a system of integer multiplicities that form matrix representations of fusion-like algebras. These multiplicities are also the combinatorial data that enable one to construct an abstract ``quantum'' algebra, whose $6j$- and $3j$-symbols contain essential information on the Operator Product Algebra of the RCFT and are part of a cell system, subject to pentagonal identities. It looks quite plausible that the classification of a wide class of RCFT amounts to a classification of ``Weak $C^*$- Hopf algebras''.Comment: 23 pages, 12 figures, LateX. To appear in MATHPHYS ODYSSEY 2001 --Integrable Models and Beyond, ed. M. Kashiwara and T. Miwa, Progress in Math., Birkhauser. References and comments adde

Supersymmetric matrix models and branched polymers

We solve a supersymmetric matrix model with a general potential. While matrix models usually describe surfaces, supersymmetry enforces a cancellation of bosonic and fermionic loops and only diagrams corresponding to so-called branched polymers survive. The eigenvalue distribution of the random matrices near the critical point is of a new kind.Comment: xx pages, Latex, no macros neede

On the Spectral Gap of a Quantum Graph

We consider the problem of finding universal bounds of "isoperimetric" or "isodiametric" type on the spectral gap of the Laplacian on a metric graph with natural boundary conditions at the vertices, in terms of various analytical and combinatorial properties of the graph: its total length, diameter, number of vertices and number of edges. We investigate which combinations of parameters are necessary to obtain non-trivial upper and lower bounds and obtain a number of sharp estimates in terms of these parameters. We also show that, in contrast to the Laplacian matrix on a combinatorial graph, no bound depending only on the diameter is possible. As a special case of our results on metric graphs, we deduce estimates for the normalised Laplacian matrix on combinatorial graphs which, surprisingly, are sometimes sharper than the ones obtained by purely combinatorial methods in the graph theoretical literature