Meander, Folding and Arch Statistics

The statistics of meander and related problems are studied as particular realizations of compact polymer chain foldings. This paper presents a general discussion of these topics, with a particular emphasis on three points: (i) the use of a direct recursive relation for building (semi) meanders (ii) the equivalence with a random matrix model (iii) the exact solution of simpler related problems, such as arch configurations or irreducible meanders.Comment: 82 pages, uuencoded, uses harvmac (l mode) and epsf, 26+7 figures include

Strings, Matrix Models, and Meanders

I briefly review the present status of bosonic strings and discretized random surfaces in D>1 which seem to be in a polymer rather than stringy phase. As an explicit example of what happens, I consider the Kazakov-Migdal model with a logarithmic potential which is exactly solvable for any D (at large D for an arbitrary potential). I discuss also the meander problem and report some new results on its representation via matrix models and the relation to the Kazakov-Migdal model. A supersymmetric matrix model is especially useful for describing the principal meanders.Comment: 12 pages, 4 Latex figures, uses espcrc2.sty Talk at the 29th Ahrenshoop Symp., Buckow, Germany, Aug.29 - Sep.2, 199

Folding Transitions of the Square-Diagonal Lattice

We address the problem of "phantom" folding of the tethered membrane modelled by the two-dimensional square lattice, with bonds on the edges and diagonals of each face. Introducing bending rigidities $K_1$ and $K_2$ for respectively long and short bonds, we derive the complete phase diagram of the model, using transfer matrix calculations. The latter displays two transition curves, one corresponding to a first order (ferromagnetic) folding transition, and the other to a continuous (anti-ferromagnetic) unfolding transition.Comment: TeX using harvmac.tex and epsf.tex, 22 pages (l mode), 17 figure

A transfer matrix approach to the enumeration of plane meanders

A closed plane meander of order $n$ is a closed self-avoiding curve intersecting an infinite line $2n$ times. Meanders are considered distinct up to any smooth deformation leaving the line fixed. We have developed an improved algorithm, based on transfer matrix methods, for the enumeration of plane meanders. While the algorithm has exponential complexity, its rate of growth is much smaller than that of previous algorithms. The algorithm is easily modified to enumerate various systems of closed meanders, semi-meanders, open meanders and many other geometries.Comment: 13 pages, 9 eps figures, to appear in J. Phys.

Meanders: Exact Asymptotics

We conjecture that meanders are governed by the gravitational version of a c=-4 two-dimensional conformal field theory, allowing for exact predictions for the meander configuration exponent \alpha=\sqrt{29}(\sqrt{29}+\sqrt{5})/12, and the semi-meander exponent {\bar\alpha}=1+\sqrt{11}(\sqrt{29}+\sqrt{5})/24. This result follows from an interpretation of meanders as pairs of fully packed loops on a random surface, described by two c=-2 free fields. The above values agree with recent numerical estimates. We generalize these results to a score of meandric numbers with various geometries and arbitrary loop fugacities.Comment: new version with note added in proo