Critical exponents of plane meanders

Meanders form a set of combinatorial problems concerned with the enumeration of self-avoiding loops crossing a line through a given number of points, n. Meanders are considered distinct up to any smooth deformation leaving the line fixed. We use a recently developed algorithm, based on transfer matrix methods, to enumerate plane meanders. This allows us to calculate the number of closed meanders up to n = 48, the number of open meanders up to n = 43, and the number of semi-meanders up to n = 45. The analysis of the series yields accurate estimates of both the critical point and critical exponent, and shows that a recent conjecture for the exact value of the semi-meander critical exponent is unlikely to be correct, while the conjectured exponent value for closed and open meanders is not inconsistent with the results from the analysis. Meanders form a set of unsolved combinatorial problems concerned with the enumeration of self-avoiding loops crossing a line through a given number of points [1]. Meanders are considered distinct up to any smooth deformation leaving the line fixed. This problem seems to date back at least to the work of Poincaré on differential geometry [2]. Since then it has from time to time been studied by mathematicians in various contexts such as the folding of a strip of stamps [3, 4] or folding of maps [5]. More recently it has been related to enumerations of ovals in planar algebraic curves [6] and the classification of 3-manifolds [7]. During the last decade or so it has received considerable attention in other areas of science. In computer science meanders are related to the sorting of Jordan sequences [8] and have been used for lower bound arguments [9]. In physics meanders are relevant to the study of compact foldings of polymers [10, 11], properties of the Temperley-Lieb algebra [12, 13], matrix models [14, 15], and models of low-dimensional gravity [16]. A closed meander of order n is a closed self-avoiding loop crossing an infinite line 2n times (see figure 1). The meandric number Mn is simply the number of such meanders distinct up to smooth transformations. Note that each meander forms a single connected loop. The number of closed meanders is expected to grow exponentially, with a sub-dominant term given by a critical exponent, Mn ∼ CR/n. The exponential growth constant R is often called the connective constant. Thus the generating function is expected to behave as e-mail: [email protected] e-mail: [email protected]