Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 214 / 3

A voltage with boundary values $u(a)$ and $u(b)$ is a function $u:V\to \mathbb{R}$ that is harmonic at every other vertex:A current flow is an antisymmetric function $i(x,y)=-i(y,x)$ satisfying Kirchhoff's node law ${\sum }_{y}i(x,y)=0$ away from its source and sink. Voltage and current are related by Ohm's law,The effective resistance is the voltage drop divided by the total current from $a$ to $b$. Equivalently, it is the voltage drop generated by a unit current flow:

Put $A=\{a,b\}$ and let ${\tau }_x^{+}=\min \{t\ge 1:X_t=x\}$. The conductance-hitting identity for an electrical network iswhere the vertices of $D$ are wired together. It follows by taking the hitting probability of $D$ before returning to $x$ as a voltage and computing its total current out of $x$.

Split the walk into successive excursions from $x$. The first excursion that hits $A$ determines whether $a$ or $b$ is hit first. Its conditional probability of hitting $a$ is at most the probability that an arbitrary excursion hits $a$, divided by the probability that it hits $A$. Hence

Let $v$ be the voltage with $v(a)=1$, $v(b)=0$, and harmonic values at every other vertex. For any other admissible $f$, write $f=v+g$, where $g(a)=g(b)=0$. Its discrete Dirichlet energy expands asDiscrete summation by parts turns the cross term intobecause $v$ is harmonic in the interior and $g$ vanishes at the boundary. Thus $v$ minimizes the energy. Under a unit voltage drop, its energy equals the total current from $a$ to $b$, namely the effective conductance $1/R_{\mathrm{eff}}(a,b)$. Therefore the Dirichlet principle gives