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

Abstract

We find the order of the variance of the growth model $X_{n+1}=X_n+X_n^{\prime }+f(X_n^{\prime \prime },X_n^{\prime \prime \prime })$, where all the variables $X_n,X_n^{\prime },X_n^{\prime \prime }$ and $X_n^{\prime \prime \prime }$ are i.i.d., $X_0$ 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 $G_n$ is of order $(2+\frac{1}{8}+\mathcal{O}(1){)}^n$.

10 pages