Variance of Resistance of "Line-Circle-Line" Graphs

Original · EN

We find the order of the variance of the growth model Xₙ₊₁ = Xₙ+X'ₙ + f(X''ₙ,X'''ₙ), where all the variables Xₙ,X'ₙ,X''ₙ and X'''ₙ are i.i.d., X₀ takes the values 1 and 2 with equal probability and f is positive, monotone non-decreasing and satisfies conditions which, roughly speaking, pertain to its first and second order partial derivatives. For an appropriate choice of f we obtain that the variance of the effective resistance between the endpoints of the "line-circle-line" graph Gₙ is of order (2 + 1/8 + (1))ⁿ.