The form of MinNet given here can take up to about n 2 steps to generate a result with n nodes; an n Log[n] procedure is known.

But in a case like (a) on page 514 —where spacetime has the structure of an exponentially growing tree—points a distance t apart typically have common ancestors just Log[t] steps back.

Quite a few were based just on changing p i Log[p i ] in the definition of entropy to a quantity vanishing for both ordered and disordered p i .

(The proof is based on having bounds for how close to zero Sum[ α i , Log[ α i ], i, j] can be for independent algebraic numbers α k .)

But for smaller e[s] one can show that Abs[m[s]]  (1 - Sinh[2 β ] -4 ) 1/8 where β can be deduced from e[s]  -(Coth[2 β ](1 + 2 EllipticK[4 Sech[2 β ] 2 Tanh[2 β ] 2 ] (-1 + 2 Tanh[2 β ] 2 )/ π )) This implies that just below the critical point e 0 = - √ 2 (which corresponds to β = Log[1 + √ 2 ]/2 ) Abs[m] ~ (e 0 - e) 1/8 , where here 1/8 is a so-called critical exponent.