Transient graph

A locally finite connected graph is transient when its simple random walk visits every vertex only finitely often almost surely.

For the walk on a weighted graph, the Green function normalized by vertex weight isSymmetry of the conductances makes $g(x,y)=g(y,x)$, and $\mathcal{E}(g(\cdot ,y),f)=f(y)$ for finitely supported $f$.

For simple random walk on ${\mathbb{Z}}^d$ with $d\ge 3$,Consequently the probability of ever hitting $x$ from the origin has the same order.

For a finite vertex set $A$, its equilibrium potential is $h_A(x)=P_x(H_A<\infty )$. It equals one on $A$, is harmonic off $A$, and belongs to the Dirichlet energy space with zero boundary at infinity.

The equilibrium measure is supported on $A$ and is given by

The capacity of a finite set is the total mass of its equilibrium measure of a finite set:Capacity is monotone under inclusion, though strict inclusion need not give strict inequality.

For a vertex $o$,

A last-exit decomposition partitions a transient path event according to the final visit to a finite set. Reversing the finite path before that visit and using detailed balance converts last-exit probabilities into hitting probabilities weighted by an equilibrium measure of a finite set.