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.


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Tony Phillips
Math Dept SUNY Stony Brook
tony at math.stonybrook.edu
January 16 2014