OFFSET
1,2
COMMENTS
Chang shows that a constant population of n individuals, with ancestors selected uniformly at random, converges in probability to a state where every individual leaves either no current ancestors or else is a common ancestor of all present individuals after k*log_2(n) generations, where k is this constant (see Theorem 2 in Chang link for precise statement).
LINKS
Joseph T. Chang, Recent common ancestors of all present-day individuals, Advances in Applied Probability Vol. 31, No. 4 (Dec., 1999), pp. 1002-1026.
James Grime and Brady Haran, EVERY baby is a ROYAL baby, Numberphile video (2019).
Douglas L. T. Rohde, Steve Olson, and Joseph T. Chang, Modelling the recent common ancestry of all living humans, Nature Vol. 431, No. 7008 (Sep. 2004), pp. 562-566.
EXAMPLE
1.769755495564801280059561457905786683522251513088978630155101689614415...
A population of 1000 is expected to have identical ancestors after around k*log_2(1000) = 17.6... generations.
A population of a million is expected to have identical ancestors after around k*log_2(10^6) = 35.2... generations.
A population of a billion is expected to have identical ancestors after around k*log_2(10^9) = 52.9... generations.
A population of a trillion is expected to have identical ancestors after around k*log_2(10^12) = 70.5... generations.
MATHEMATICA
RealDigits[1 - Log[2]/Log[-ProductLog[-2/E^2]], 10, 120][[1]] (* Amiram Eldar, Jun 27 2023 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Charles R Greathouse IV, May 07 2019
STATUS
approved