Harmonic functions on planar and almost planar graphs and ...

570 I. Benjamini, O. Schramm2. Notations and terminologyLet G =(V,E) be a graph. For convenience, we usually only consider graphswith no loops or multiple edges (but the results do apply to multigraphs). The setof vertices incident with an edge e will be denoted ∂e; this is always a subset ofV that contains two vertices. We sometimes use {v, u} to denote the edge withendpoints v, u.Initially the graph G is unoriented, but for notational reasons we also considerdirected edges. When {v, u} ∈E, we use [v, u] to denote the directed edge fromv to u. The set of all direceted edges will usually be denoted −→ E ; −→ E = {[v, u] :{v, u} ∈E}.The graphs we shall consider will be connected and locally finite. The lattermeans that the number of edges incident with any particular vertex is finite.Given any vertex v ∈ V , the collection of all edges of the form [v, u] whichare in −→ E will be denoted −→ E (v). The valence, ordegree, of a vertex v is just thecardinality of −→ E (v). G has bounded valence, if there is a finite upper bound forthe degrees of its vertices.Let f : V → R be some function. Then df is the function df : −→ E → Rdefined bydf ([v, u]) = f (u) − f (v).We also define the gradient of f to be equal to df ,∇f (e) =df (e).(The reason for the multiplicity of notation should become clear when we introducethe gradient with respect to a metric on G.)A function j : −→ E → R is a flow on G if it satisfiesj ([u,v]) = −j ([v, u])for every {v, u} ∈E. For example, for any f : V → R, df is a flow. Thedivergence of a flow j is the function div j : V → R, defined bydiv j (v) =∑j(e).If div j = 0, then j is divergence free.For an f : V → R we sete∈ −→ E(v)△f = div ∇f ,then △f : V → R is known as the discrete laplacian of f .If△f = 0, then f isharmonic, while if △f = 0 on a subset V ′ ⊂ V , we say that f is harmonic in V ′ .Equivalently, f is harmonic iff its value at any v ∈ V is equal to the average ofthe values at the neighbors of v.