In this subsection, we recall a few well-known properties of the DGFF that are valid on any finite graph. Let (V, E) be a finite graph, and letV∂⊂V be a non-empty set of vertices. Let h denote a sample from the DGFF with boundary values given by some functionh∂:V∂!R.
When f is a function on V, the discrete gradient ∇f is the function on the set of ordered pairs (v, u) such that {v, u}∈E defined by ∇f((u, v))=f(v)−f(u). When defining the norm of the gradient we sum over undirected edges, i.e., we write
k∇f(v)k2= X
{u,v}∈E
(f(v)−f(u))2. (2.1)
Thus, the probability density ofhis proportional toe−k∇h(v)k2/2. Therefore, whenh∂=0 on V∂, h is a standard Gaussian with respect to the norm k∇hk on Ω. The (discrete) Dirichlet inner product that defines this norm can be written
(f, g)∇= X
{u,v}∈E
(f(v)−f(u))(g(v)−g(u)). (2.2)
Now write ∆f(v)=P
u∼v(f(u)−f(v)), where the sum is over all neighborsuof v.
By expanding and rearranging the summands in (2.2), we find
(f, g)∇=−(∆f, g). (2.3)
LetV0⊂VD. We claim that the vector space of functions f:V!Rthat are zero on V\V0, and the vector space of functionsf:V!Rthat are discrete-harmonic on V0 (i.e.,
∆f=0 onV0) are orthogonal to each other with respect to the inner product (·,·)∇, and together they span RV. This basic observation will be used frequently below. Indeed, that they are orthogonal follows immediately from (2.3), and a dimension count now shows that the two spaces together spanRV.
The following consequence of this orthogonality property will be used below. Let V0⊂V satisfyV0⊃V∂ and let h0 denote the function that is discrete-harmonic in V\V0
and equal to h in V0. Then h−h0 and h0 are independent random variables, because h=h0+(h−h0) is the corresponding orthogonal decomposition ofh. It also follows that h−h0is the DGFF inV\V0 with zero boundary values onV0. (2.4) Observe that the Markov property and the effect of boundary conditions that were men-tioned in§1.2 are immediate consequences of (2.4). An additional useful consequence is that the law ofh0is proportional to e−(h0,h0)2∇/2 times the Lebesgue measure onRV0.
We now derive a useful well-known expression for the expectation ofh(v)h(u):
E[h(v)h(u)]−E[h(v)]E[h(u)] =G(u, v)
deg(v), (2.5)
whereG(u, v) is the expected number of visits tov by a simple random walk started at ubefore it hits V∂ and deg(v) is the degree ofv; that is, the number of edges incident with it. (The function G is known as the Green function.) As we have noted in the introduction, his the sum of the discrete-harmonic extension of h∂ and a DGFF with zero boundary values. It therefore suffices to prove (2.5) in the case whereh∂=0 onV∂. In this case, E[h(v)]=0=E[h(u)]. Setting Gv(u)=G(u, v), we observe (or recall) that
∆Gv(u)=−deg(v)1v(u). Thus,
h(v) = (h,1v) =−(h,∆Gv) deg(v)
(2.3)
= (h, Gv)∇
deg(v) .
If X is a standard Gaussian in Rn, and x, y∈Rn, then E[(X·x)(X·y)]=x·y. Conse-quently, whenh∂=0, we have
E[h(v)h(u)] =E[(h, Gv)∇(h, Gu)∇]
deg(v) deg(u) = (Gv, Gu)∇
deg(v) deg(u)
= −(Gv,∆Gu)
deg(v) deg(u)=(Gv,1u)
deg(v) =G(u, v) deg(v). This proves (2.5).
2.2. Some assumptions and notation
We will make frequent use of the following notation and assumptions:
(h) A bounded domain (non-empty, open, connected set) D⊂R2 whose boundary
∂D is a subgraph of TG is fixed. The set of vertices of TG inD is denoted byVD and V∂ denotes the set of vertices in ∂D. A constant ¯Λ>0 is fixed, as well as a function h∂:V∂!Rsatisfyingkh∂k∞6Λ. The DGFF on¯ D with boundary values given byh∂ is denoted byh. Also setV=VD=VD∪V∂.
We denote by TG∗the hexagonal grid which is dual to the triangular lattice TG—so that each hexagonal face of TG∗is centered at a vertex of TG. Generally, a TG∗-hexagon will mean a closed hexagonal face of TG∗. Denote byBRthe union of all TG∗-hexagons that intersect the ballB(0, R).
Sometimes, in addition to (h) we will need to assume:
(D) The domainD is simply connected, and (to avoid minor but annoying triviali-ties)∂D is a simple closed curve. We fix two distinct midpoints of TG-edgesx∂ andy∂
on ∂D. Let the counterclockwise (respectively, clockwise) arc of ∂D from x∂ to y∂ be denoted by∂+ (respectively,∂−).
IfH⊂D is a TG∗-hexagon, we writeh(H) as a shorthand for the value ofhon the center ofH (which is a vertex of TG). Assuming (h) and (D), letD+denote the union of all TG∗-hexagons contained in D where h is positive together with the intersection of Dwith TG∗-hexagons centered at vertices in ∂+. Let D− be the closure of D\D+ (which almost surely consists of TG∗-hexagons inDwhereh<0 and the intersection ofD with TG∗-hexagons whose center is in∂−). Then∂D−∩∂D+ necessarily consists of the interface we previously calledγ, and a collection of disjoint simple closed paths. We use the terminterface (orzero-height interface) to describe a simple (or simple closed) path in∂D−∩∂D+oriented so thatD+is on its right (that is, oriented clockwise aroundD+).
Throughout, the notationO(s) represents any quantityf such that|f|6Csfor some absolute constantC. We use the notationOΛ¯(s) if the constant also depends on ¯Λ. When introducing a constant c, we often write c=c(a, b) as shorthand to indicate that c may depend onaandb.
2.3. Simple random walk background
We need to recall a very useful property of the discrete-harmonic measure of simple random walk.
Lemma 2.1. (Hit near) Let v be a vertex of the grid TG,and let H be a connected subgraph of TG. Set d=diamH. The probability that a simple random walk on TG started from v exits the ball B(v, d)before hitting H is at most c(dist(v, H)/d)ζ1,where c and ζ1∈(0,1) are absolute constants.
Likewise,the same bound applies to the probability that a simple random walk started at some vertex outside B(v, d)will hit B(v,dist(v, H))before H.
In fact, we may takeζ1=12. The continuous version of this statement is known as the Beurling projection theorem (the extremal case is whenH is a line segment). The above statement can probably be deduced from the discrete Beurling theorem as given
in [L1, Theorem 2.5.2], though the setting there is slightly different. In any case, since we do not require any particular value forζ1, the lemma is rather easily proved directly (see [Sch, Lemma 2.1]).
We will also use the (well-known) discrete Harnack principle, in the following form.
Lemma 2.2. (Harnack principle) Let D, V, V∂ and VD be as in (h), let v, u∈VD, and let f:V!R be discrete-harmonic in VD.
(1) If v and uare neighbors, then
|f(v)−f(u)|6O(1) kfk∞ dist(v, ∂D).
(2) If we assume that f>0 and that there is a path of length ` from v to u whose minimal distance to ∂D is %, then
f(u)>f(v) exp
−O(`+%)
%
.
The proof of statement (1) can be obtained by noting thatf(v) is the expected value off evaluated at the first hitting point onV∂ of a random walk started atv, and observing that a random walk started atv and a random walk started atumay be coupled so that they meet before reaching a distance ofr:=dist(v, ∂D) with probability 1−O(1/r) and walk together after they meet. (See also the more general [LSW4, Lemma 6.2], for example.)
To prove statement (2), letW:={w∈V:f(w)>f(v)}, and observe that the maximum principle implies thatW contains a path fromv to∂D. If we assume that`612%, then a random walk started atuhas probability bounded away from zero to hitW before∂D, which by the optional sampling theorem implies (2) in this case. The case `>12% now follows by induction ond2`/%e.
We also need the following well-known estimate on the Green functionG(u, v).
Lemma 2.3. Let D, VD and V be as in (h), let u, v∈VD, and let ε>0. Set r:=
dist(u, ∂D),and suppose that within distance r/εfrom uthere is a connected component of ∂D of diameter at least εr. If |u−v|623r, say, then GD(v, u) (the expected number of visits to uby a simple random walk starting at v before hitting ∂D)satisfies
GD(v, u) = exp(Oε(1)) log r+1
|u−v|+1.
The probabilityHD(v, u) that a random walk started fromv hits ubefore ∂Dcan be expressed asGD(v, u)/GD(u, u) and hence by the lemma
HD(v, u) = exp(Oε(1))
1−log(|u−v|+1) log(r+1)
. (2.6)
Let pR denote the probability that a simple random walk started at 0 does not return to 0 before exiting the ballB(0, R). We now show that the lemma follows from the well-known estimate
pR=exp(O(1))
log(R+2). (2.7)
In the setting of the lemma, consider some vertexw∈VDsuch that|w−u|>r. It is easy to see that with probability bounded away from zero (by a function ofε), a simple random walk started fromwwill hit∂Dbeforeu. Hence,q:=min{1−HD(w, u):w∈VD\B(u, r)}
is bounded away from 0 by a positive function ofε. Clearly, 1
qpr>GD(u, u)> 1 pr
.
This proves the caseu=v of the lemma from (2.7). Now start a simple random walk at a vertexv6=usatisfying|v−u|623r, and let the walk stop when it hits∂D. It is easy to see that the probability that the walk visitsv after exiting the ballB v,14|v−u|
, say, is within a bounded factor from HD(v, u). Therefore, the expected number of visits tov after exitingB v,14|v−u|
is within a bounded factor ofGD(v, u). But the former is the same asGD(v, v)−GB(v,|v−u|/4)(v, v). The lemma follows.
We have not found a reference proving (2.7) in a way that generalizes to the setting of Theorem 1.4, though the result is well known. In fact, it easily follows from Rayleigh’s method, as explained in [DoSn,§2.2]. In that book, the goal is to show thatpR!0, as R!∞, for lattices in the plane but not in R3. However, the method easily yields the quantitative bounds (2.7).
3. The height gap in the discrete setting