Through Mazes to Mathematics

## Recent developments - September 2000

I am indebted to the authors who sent me preprints of their work, and also to Steven Finch, who recently filled me in on details I had missed.

I. Experimental. The calculation of maze numbers has been carried much further by Iwan Jensen and Anthony J. Guttmann of the University of Melbourne, using an algorithm based on transfer matrix methods. Here are the numbers they published in Critical exponents of plane meanders.

Table 1. The number Mn  of connected closed meanders with 2n crossings.

 1 1 2 2 3 8 4 42 5 262 6 1 828 7 13 820 8 110 954 9 933 458 10 8 152 860 11 73 424 650 12 678 390 116 13 6 405 031 050 14 61 606 881 612 15 602 188 541 928 16 5 969 806 669 034 17 59 923 200 729 046 18 608 188 709 574 124 19 6 234 277 838 531 806 20 64 477 712 119 584 604 21 672 265 814 872 772 972 22 7 060 941 974 458 061 392 23 74 661 728 661 167 809 752 24 794 337 831 754 564 188 184

The number of closed meanders is expected to grow exponentially, with a sub-dominant term given by a critical exponent,  Mn ~ C R2n/nα The exponential growth constant R is often called the connective constant", while α is the coefficient exponent."

Using these and other data, Jensen and Guttmann estimate the constants as R = 3.501 837(3) and alpha = 3.4208(6).

II. Theoretical. P. Di Francesco, O. Golinelli and E. Guitter, of the Service de Physique Théorique at Saclay have a sequence of papers culminating in Meanders: exact asymptotics. There they propose a model from conformal field theory ("the gravitational version of a c=-4 two-dimensional conformal field theory") which allows them to conjecture an exact limit for the Meander coefficient exponent. Their number 291/2[291/2 + 51/2]/12 is in agreement with the Australian team's estimates.

M.H. Albert (Computer Science, University of Otago) and M.S. Patterson (Computer Science, University of Warwick) have published Bounds for the Growth Rate of Meander Numbers (2004). They define Mn to be the number of meanders which cross the vertical axis at 2n points, and seek upper and lower bounds on the exponential part of the asymptotic form. They observe that M = limn-->∞M n1/n exists and show that 11.380 ≤ M ≤ 12.901.