A370088 Decimal expansion of the two-dimensional backbone constant.

3, 5, 6, 6, 6, 6, 8, 3, 6, 7, 1, 2, 8, 8, 9, 5, 8, 2, 8, 3, 7, 3, 0, 7, 3, 8, 1, 0, 0, 1, 2, 6, 6, 2, 6, 9, 9, 0, 3, 8, 7, 0, 1, 5, 3, 4, 0, 7, 6, 2, 4, 4, 1, 3, 9, 9, 0, 6, 0, 9, 7, 3, 7, 6, 3, 7, 3, 6, 1, 3, 8, 4, 2, 0, 8, 8, 8, 5, 5, 4, 8, 5, 1, 9, 6, 7, 2

Offset: 0

Author

Charles R Greathouse IV, Feb 08 2024

Keywords

• cons
• nonn

Comments

This constant is the negative of the exponent of the growth rate of the probability of Bernoulli percolation on the 2-dimensional triangular lattice at criticality (p = 1/2). It is transcendental (Theorem 1.2 in Nolin, Qian, Sun, & Zhuang).

Examples

			0.35666683671288958283730738100126626990387015340762441399060973763736138420....
		

Links

• Pierre Nolin, Wei Qian, Xin Sun, and Zijie Zhuang, Backbone exponent for two-dimensional percolation, arXiv preprint (2023). arXiv:2309.05050 [math.PR], 2023-2024.
• Leila Sloman, Maze proof establishes a 'backbone' for statistical mechanics, Quanta Magazine (2024).
• Index entries for transcendental numbers

Crossrefs

Cf. A003197, A003198, A003202, A006809.

Programs

PARI

t=sqrt(3)/4; u=2*Pi/3; solve(x=.3,.4, my(s=sqrt(12*x+1)); sin(s*u)+s*t)

Formula

This is the unique constant 1/4 < x < 2/3 with sqrt(36*x+3)/4 + sin(2*Pi*sqrt(12*x+1)/3) = 0.