Matrix model combinatorics: applications to folding and coloring

We present a detailed study of the combinatorial interpretation of matrix integrals, including the examples of tessellations of arbitrary genera, and loop models on random surfaces. After reviewing their methods of solution, we apply these to the study of various folding problems arising from physics, including: the meander (or polymer folding) problem ``enumeration of all topologically inequivalent closed non-intersecting plane curves intersecting a line through a given number of points" and a fluid membrane folding problem reformulated as that of ``enumerating all vertex-tricolored triangulations of arbitrary genus, with given numbers of vertices of either color".

[1]  James Brown,et al.  Strings , 1930, The Python Book.

[2]  W. T. Tutte.  A Census of Planar Maps , 1963, Canadian Journal of Mathematics.

[3]  W. F. Lunnon.  A map-folding problem , 1968 .

[4]  C. Itzykson,et al.  The planar approximation. II , 1980 .

[5]  J. Duistermaat,et al.  On the variation in the cohomology of the symplectic form of the reduced phase space , 1982 .

[6]  Kurt Mehlhorn,et al.  Sorting Jordan Sequences in Linear Time Using Level-Linked Search Trees , 1986, Inf. Control..

[7]  F. David,et al.  Exact partition functions and correlation functions of multiple Hamiltonian walks on the Manhattan lattice , 1988 .

[8]  Lawrence J. Smolinsky,et al.  A combinatorial matrix in $3$-manifold theory. , 1991 .

[9]  C. Itzykson,et al.  A Generating Function for Fatgraphs , 1992, hep-th/9212108.

[10]  P. Di Francesco,et al.  2D gravity and random matrices , 1993 .

[11]  A. K. Zvonkin,et al.  Plane and Projective Meanders , 1993, Theor. Comput. Sci..

[12]  Folding transition of the triangular lattice. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[13]  E. Guitter,et al.  Meander, Folding and Arch Statistics , 1995 .

[14]  Supersymmetric Matrix Models and the Meander Problem , 1996, hep-th/9601139.

[15]  Hermitian matrix model with plaquette interaction , 1996, hep-th/9605013.

[16]  P. Di Francesco,et al.  Meander Determinants , 1996, hep-th/9612026.

[17]  Matthias Staudacher,et al.  Character expansion methods for matrix models of dually weighted graphs , 1995, hep-th/9502132.

[18]  Quantum Integrable Systems and Elliptic Solutions of Classical Discrete Nonlinear Equations , 1996, hep-th/9604080.

[19]  E. Guitter,et al.  Meanders: A direct enumeration approach , 1996, hep-th/9607039.

[20]  E. Guitter,et al.  Meanders and the Temperley-Lieb algebra , 1997 .

[21]  P. Francesco.  SU(N) meander determinants , 1997, hep-th/9702181.

[22]  Transition from the Compact to the Dense Phase of Two-Dimensional Polymers , 1998, cond-mat/9811085.

[23]  Field theory of compact polymers on the square lattice , 1998, cond-mat/9804048.

[24]  P. Francesco.  Folding the square-diagonal lattice , 1998, cond-mat/9803051.

[25]  Folding transitions of the square-diagonal lattice , 1998, cond-mat/9804073.

[26]  E. Guitter,et al.  Coloring random triangulations , 1998 .

[27]  G. Bonnet,et al.  The Potts-q random matrix model: loop equations, critical exponents, and rational case , 1999 .

[28]  TWO-DIMENSIONAL COPOLYMERS AND EXACT CONFORMAL MULTIFRACTALITY , 1998, cond-mat/9812439.

[29]  P. Di Francesco,et al.  Folding and coloring problems in mathematics and physics , 2000 .

[30]  A Monte-Carlo study of meanders , 1999, cond-mat/9906329.