Discrete Gaussian Free Field via Hadamard's formula

DISCRETE GA USSIAN FREE FIELD VIA HAD AMARD’S F ORMULA HAAKAN HEDENMALM, P A VEL MOZOL Y AK O, AND DANIIL P ANOV Abstract. W e presen t a no v el wa y of constructing the Gaussian F ree Field on a w eigh ted graph via a dynamical expansion of the Green function along an expanding collection of subgraphs. Along the w a y w e obtain the discrete analogue of the classical Hadamard v ariational form ula regarding the v ariation of the Green function under infinitesimal v ariations of the domain. In order to dev elop necessary mac hinery we construct expanding bases of the naturally asso ciated energy spaces. An interesting observ ation is that both our discrete Hadamard v ariational formula and the related construction of the discrete Gaussian F ree Field are completely dimension-free and do not require smoothness of an y kind. The graph mo del contains geometric information via the edges whic h supply the discrete top ological information, and by conductances which give metric information. Going to a con tinuum limit, w e w ould then obtain con tinuous v ersion of the Hadamard v ariational formula and the asso ciated Hadamard op erator in e.g. fractal geometries of arbitrary dimension. 1. Introduction 1.1. The DGFF process. The Discrete Gaussian F ree Field (DGFF) on a graph Γ is an analogue of the classical Gaussian F ree Field (GFF), whic h app ears naturally as a discrete approximation of the latter, while b eing a p opular ob ject of research in its o wn right (see [DLP], [B]). It can b e defined in man y w a ys, one of these is a random linear combination of an orthonormal basis { ψ k } of the discrete Sob olev space H 1 0 ( U ) by Ψ = P k ξ k ψ k , where ξ k is a sequence of indep enden t standard real-v alued Gaussian v ariables. The cov ariance structure of the discrete sto c hastic field Ψ is given by the Green function of the subgraph U . 1.2. Gro wing the DGFF along a discrete foliation. In 2014, Hedenmalm and Nieminen [HN] studied the GFF in the contin uous setting of a planar domain. They found an interpretation of the classical v ariational form ula of Hadamard, which giv es the c hange of the Green function under a p erturbation of the domain. The interpretation giv es a recip e for ho w to build the GFF proc ess along a lamination of the domain b y contin uously adding indep enden t White Noise (WN) contributions, extended harmonically to the interior. The ob jectiv e of this note is to construct a similar scheme in the discrete context on a fairly general graph – to grow the DGFF in a lay er-b y-la y er fashion by adding (weigh ted) harmonic extensions of the White Noise Field (WNF) based on a discrete v ersion of Hadamard’s formula. Since our construction holds under very general assumptions on the graph, the conclusion is that w e may infer that a more general theorem should b e v alid in any reasonable con tin uum limit of our graphs, suc h as on surfaces or manifolds (ev en fractal) in higher dimensions. 1.3. Structure of the pap er. The pap er is organized as follows. In Section 2, we collect the nec- essary background material regarding definitions and related facts. Then, in Section 3, we introduce the DGFF and WNF. W e also outline the general plan of the DGFF gro wth algorithm on graphs. The precise algorithmic construction is presen ted in detail in Section 4, where, in particular, w e ob- tain the discrete Hadamard formula (Theorem 4.2). Our main result, stated in Theorem 5.1, asserts that the DGFF can b e view ed as the result of adding independent discrete WNF harmonic wa ves 2020 Mathematics Subje ct Classific ation. 31C20, 60G60. The work is supported by Ministry of Science and Higher Education of the Russian F ederation under agreement No. 075-15-2025-013. 1 2 H. HEDENMALM, P . MOZOL Y AKO, AND D. P ANOV along the c hosen gro wth pattern, analogously to what happ ens in the contin uous setting. The details are presented in Section 5. Finally , in Section 6, we discuss some prop erties of the DGFF that are immediate from the construction. 2. Discrete setting and not a tion 2.1. Purp ose of this section. Here, we introduce the discrete setting we are going to work with and the related notation. 2.2. Graphs with conductances. Let Γ = ( V , E ) b e a connected lo cally finite graph. Here V stands for the collection of v ertices, and E for the collection of edges. W e will frequen tly write V = V (Γ) and E = E (Γ) to indicate that V and E are asso ciated w ith the giv en graph Γ. Edges connect a pair of vertices, and w e write e = ( x, y ) for the edge connecting vertex x with the differen t v ertex y . F or instance, we assume that for an y x ∈ V : ( x, x ) / ∈ E and for any edge e = ( x, y ) there is a reverse edge ( y , x ) ∈ E , denoted by ( − e ). In other words, the set E of edges is symmetric and irreflexiv e. W e will write x ∼ y to express that ( x, y ) is in the edge set E , and think of x and y as neighb ors . If there are comp eting graphs to b e concerned with, we use a subscript to indicate with resp ect to which graph the given relation is tak en, e.g. x ∼ Γ y . Given an edge e ∈ E , w e express the v ertices that it connects b y e − and e + , in the sense that if e = ( x, y ) then x = e − and y = e + . Let c b e a c onductanc e [LP, Chapter 2], that is, a p ositive symmetric w eight on the edges of Γ, and for x ∈ V , let π ( x ) = P y : y ∼ x c ( x, y ) b e the stationary distribution for the corresp onding random w alk. 2.3. Hilb ert spaces on vertices and on edges. F or a finite subset U ⊂ V of v ertices, we define the real Hilb ert space of functions on vertices ℓ 2 ( U ) := { f : V → R , supp f ⊂ U } with the inner pro duct ⟨ f , g ⟩ U := X x ∈ U f ( x ) g ( x ) . Elemen ts f ∈ ℓ 2 ( U ) are tacitly assumed extended to all v ertices in V b y declaring that f ( x ) = 0 for all x ∈ V \ U . W e shall need also the corresp onding concept of Hilb ert spaces on the edges. So, for a finite symmetric set F ⊂ E of edges (i.e., e ∈ F if and only if ( − e ) ∈ F ), we define the real Hilb ert space of an tisymmetric functions on those edges ℓ 2 − ( F ) :=  ϕ : → R : ∀ e ∈ F : ϕ ( e ) = − ϕ ( − e )  , supplied with the inner pro duct ⟨ ϕ, ψ ⟩ F − := 1 2 X e ∈ F ϕ ( e ) ψ ( e ) . As with the vertices, w e let the elements ϕ ∈ ℓ 2 − ( F ) extend to all the edges in E by declaring that ϕ ( e ) = 0 if e ∈ E \ F . 2.4. W eigh ted b oundary and cob oundary op erators. W e no w define the (weigh ted coboundary) op erator d : d f ( x, y ) = p c ( x, y )( f ( x ) − f ( y )) . As defined, the coboundary op erator d tak es functions defined on the v ertices and pro duces antisym- metric functions defined on the edges. W e also define the asso ciated (weigh ted b oundary) op erator d ∗ : d ∗ ϕ ( x ) = X e ∈ E : e − = x p c ( e ) ϕ ( e ) . F or a finite subset U ⊂ V , w e denote b y E U the collection of edges in the giv en graph Γ that ha ve at least one endp oin t in U . W e then immediately obtain the follo wing statemen t. DGFF VIA HADAMARD 3 Prop osition 2.1. L et f ∈ ℓ 2 ( U ) , ϕ ∈ ℓ 2 − ( E U ) . Then: ⟨ f , d ∗ ϕ ⟩ U = ⟨ d f , ϕ ⟩ E − U . Mor e over, if g is any function on V , then d ∗ d g ( x ) = X y : y ∼ x c ( x, y )( g ( x ) − g ( y )) . Next, we fix an arbitrary finite nonempty subset U ⊂ V . W e define the discr ete Sob olev sp ac e H 1 0 ( U ) = { f : V → R : supp f ⊂ U } , where the supp ort has the standard discrete meaning: supp f := { x ∈ V : f ( x )  = 0 } . W e endo w H 1 0 ( U ) with the inner pro duct structure giv en by ⟨ f , g ⟩ ∇ ,U := ⟨ d f , d g ⟩ E − U = ⟨ d ∗ d f , g ⟩ U . The asso ciated Hilb ert space norm ∥ f ∥ 2 ∇ ,U := ⟨ d f , d f ⟩ ∇ ,U = ⟨ d f , d f ⟩ E − U then extends naturally to a seminorm on all functions f : V → R . It is often referred to as the discr ete Dirichlet ener gy on U . 2.5. The discrete Laplacian. It is natural to in tro duce the op erator ∆ U : ℓ 2 ( U ) → ℓ 2 ( U ): ∆ U f ( x ) := 1 U d ∗ d f ( x ) , where 1 U denotes the characteristic function of the subset U ⊂ V . As a matter of definition, we say that a function f : V → R is discr ete harmonic on U if ∆ U f = 0 holds. Such functions are critical p oin ts of the discrete Diric hlet energy on U under additive perturbations using functions from H 1 0 ( U ). According to the properties ( a ) and ( b ) of the following prop osition, the op erator ∆ U defines a self-adjoin t op erator with nonnegativ e eigen v alues on the finite-dimensional space ℓ 2 ( U ), in which case it has a naturally defines square ro ot ( ∆ U ) 1 2 with nonnegative eigenv alues. Hence part ( c ) of the same prop osition also mak es sense. Prop osition 2.2. The fol lowing pr op erties hold for e ach finite subset U ⊂ V : ( a ) ⟨ ∆ U f , g ⟩ U = ⟨ f , ∆ U g ⟩ U for any f , g ∈ ℓ 2 ( U ) . ( b ) Al l eigenvalues of the op er ator ∆ U ar e nonne gative. ( c ) The op er ator ( ∆ U ) 1 2 defines an isometry H 1 0 ( U ) → ℓ 2 ( U ) . Pr o of. W e note that ⟨ ∆ U f , g ⟩ U = ⟨ d ∗ d f , g ⟩ U = ⟨ d f , d g ⟩ E − U = ⟨ f , d ∗ d g ⟩ U = ⟨ f , ∆ U g ⟩ U , whic h sho ws that the assertion ( a ) holds. Next, let λ b e an eigen v alue of the self-adjoint op erator ∆ U and let f ∈ ℓ 2 ( U ) b e a corresp onding eigenfunction. Since λ ⟨ f , f ⟩ U = ⟨ ∆ U f , f ⟩ U = ⟨ d ∗ d f , f ⟩ U = ⟨ d f , d f ⟩ E − U = ⟨ f , f ⟩ ∇ ,U ≥ 0 , it follows that λ ≥ 0, and assertion ( b ) follows as well. Finally , given any t wo functions f , g ∈ H 1 0 ( U ) w e obtain that  ( ∆ U ) 1 2 f , ( ∆ U ) 1 2 g  U = ⟨ ∆ U f , g ⟩ U = ⟨ d ∗ d f , g ⟩ U = ⟨ f , g ⟩ H 1 0 ( U ) . It is now immediate that the op erator ( ∆ U ) 1 2 constitutes an isometry H 1 0 ( U ) → ℓ 2 ( U ), and prop erty ( c ) follows as well. □ 4 H. HEDENMALM, P . MOZOL Y AKO, AND D. P ANOV 2.6. The discrete Green function. F or y ∈ U , we define the discr ete Gr e en function G U ( · , y ) as the solution to the follo wing system: ( d ∗ d G U ( · , y ) = π ( y ) δ y on U, G U ( x, y ) = 0 , x ∈ V \ U, where w e recall that π ( x ) = P y : y ∼ x c ( y , x ) stands for the stationary distribution for the random w alk. Here, of course, δ y ( x ) is understoo d in the Kronec k er sense, so that δ y ( x ) = 0 of x  = y while δ y ( y ) = 1. W e also declare that G U ( x, y ) = 0 holds for y ∈ V \ U, x ∈ V . W e refer to [LP, Chapter 2] on the existence and uniqueness of suc h a function. The normalized Green op erator ˜ G U : ℓ 2 ( U ) → ℓ 2 ( U ) is giv en by ˜ G U f ( x ) = X y ∈ U ˜ G U ( x, y ) f ( y ) , where ˜ G U ( x, y ) = G U ( x, y ) π ( y ) . Then, b y definition, ∆ U ˜ G U = id ℓ 2 ( U ) is the iden tit y op erator on ℓ 2 ( U ), so that ˜ G U constitutes a righ t inv erse to ∆ U . T aking adjoin ts, we obtain (2.1) ˜ G ∗ U ∆ U = ˜ G ∗ U ∆ ∗ U = id ℓ 2 ( U ) , since ∆ ∗ U = ∆ U holds in accordance with Prop osition 2.2. Then, m ultiplying by ˜ G U from the right, w e find that ˜ G ∗ U = ˜ G ∗ U ∆ U ˜ G U = ˜ G ∗ U ∆ ∗ U ˜ G U = ˜ G U , making ˜ G U self-adjoin t as an op erator on ℓ 2 ( U ). This prop erty amounts to the following symmetry pr op erty of the Green function G U : (2.2) π ( x ) G U ( x, y ) = π ( y ) G U ( y , x ) , x, y ∈ V . W e also see from (2.1) that ˜ G U is also a left in verse to ∆ U on ℓ 2 ( U ), so that ˜ G U = ∆ − 1 U as op erators on ℓ 2 ( U ). 2.7. The discrete P oisson k ernel. In addition to the discrete Green function G U , we shall need the asso ciated discrete Poisson k ernel as well. This is p erhaps less canonical than the concept of the Green function, but w e ma y pro ceed as follo ws. F or a nonempty subset W ⊂ U , we define the discr ete Poisson kernel P U,W ( · , y ), for a giv en p oint y ∈ W , as the solution to the following system: ( d ∗ d P U,W ( · , y ) = 0 on U \ W , P U,W ( x, y ) = δ y ( x ) , x ∈ W . Moreo v er, for x ∈ V \ U , we declare that P U,W ( x, y ) = 0. Asso ciated with the discrete Poisson kernel, w e hav e the Poisson extension op er ator P U,W : ℓ 2 ( W ) → ℓ 2 ( U ) ⊂ ℓ 2 ( V ) giv en by the formula P U,W f ( x ) = X y ∈ W P U,W ( x, y ) f ( y ) , x ∈ V . While P U,W f automatically v anishes on V \ U , w e then ha v e that P U,W f = f on W while P U,W f is harmonic on U \ W . As suc h, it is unique. Regarding existence and uniqueness issues, w e again refer to [LP, Chapter 2]. DGFF VIA HADAMARD 5 2.8. F oliating the graph Γ . W e turn to the con text we are going to work with to dev elop the Hadamard formula. First we need the notion of a sub gr aph . Definition 2.1. W e sa y that γ = ( V ( γ ) , E ( γ )) is a sub gr aph of Γ = ( V , E ) if V ( γ ) ⊂ V and E ( γ ) ⊂ E , while if ( x, y ) ∈ E ( γ ), then necessarily x, y ∈ V ( γ ). Moreov er, we say that γ is an exact sub gr aph of Γ if it is a subgraph, and whenever x, y ∈ V ( γ ) and ( x, y ) ∈ E , then ( x, y ) ∈ E ( γ ). Definition 2.2. A sequence { γ n } n , indexed by n ∈ Z ≥ 0 and consisting of non-empt y finite subgraphs of Γ, is called a discr ete foliation , if it has the follo wing prop erties: ( a ) F or each n , γ n is an exact subgraph. ( b ) F or m  = n , the asso ciated sets of v ertices V ( γ m ) and V ( γ n ) are disjoin t. ( c ) If x ∈ V ( γ 0 ) and y ∼ Γ x , then y ∈ V ( γ 0 ) ∪ V ( γ 1 ). ( d ) F or n = 1 , 2 , 3 , . . . : If x ∈ V ( γ n ) and y ∼ Γ x , then y ∈ V ( γ n − 1 ) ∪ V ( γ n ) ∪ V ( γ n +1 ). ( e ) W e hav e that [ n V ( γ n ) = V (Γ) . In the con text of the abov e definition, w e refer to the individual subgraphs γ n as foliating layers . Definition 2.3. Giv en a discrete foliation { γ n } n indexed by n ∈ Z ≥ 0 , we define the associated foliation gr owth cluster Γ n of Γ as the union of all preceding foliating la y ers γ k , with k = 0 , . . . , n , of the given discrete foliation, in the following sense: the vertices of Γ n are V (Γ n ) = [ 0 ≤ k ≤ n V ( γ k ) , while the set of edges E (Γ n ) consists of all the edges in Γ b et w een an y tw o vertices in V (Γ n ). 2.9. Simplification of the notational conv entions for discrete foliations. As a matter of con v enience, we prefer to write ℓ 2 (Γ n ) and H 1 0 (Γ n ) in place of ℓ 2 ( V (Γ n )) and H 1 0 ( V (Γ n )), resp ectiv ely . Analogously , w e write ∆ n for the discrete Laplacian ∆ V (Γ n ) , G n for the discrete Green function G V (Γ n ) while ˜ G n stands for the discrete normalized Green function ˜ G V (Γ n ) , and P n denotes the discrete Poisson kernel P V (Γ n ) ,V ( γ n ) . 3. Or thonormal bases in H 1 0 : constr uction of the folia tion 3.1. Aim of the section. The main purp ose of this note is the construction of a certain op erator whic h pro duces the GFF (Gaussian F ree Field) out of the harmonic extension of the WNF (White Noise Field). The most straightforw ard w a y to do so is to provide an isometry that maps an or- thonormal basis in ℓ 2 on to one in H 1 0 . In [HN] this w as carried out via the so-called Hadamar d op er ator , whic h, in turn, was extracted from the Hadamar d variational formula itself. In the discrete setting the corresp onding form ula is not readily a v ailable, nor can it b e immediately obtained b y just discretizing the corresp onding contin uous expression, so, in fact, w e need to run our argumen t bac kw ards. In other words, we assume that such a discrete isometric op erator do es exist and satisfies some natural prop erties relev ant to our setting, and then we try to deduce what it should lo ok like. Along the w ay , w e also reverse-engineer the discrete v ersion of the Hadamard v ariational formula. 3.2. The GFF and WNF pro cesses. Before we pro ceed to the actual construction, let us recall what are the discrete WNF (White Noise Field) and GFF 0 (Gaussian F ree Field with v anishing b oundary v alues) pro cesses. Let Γ U b e a finite connected subgraph of Γ, with the v ertex set U ⊂ V . W e denote the discrete WNF on the subgraph Γ U b y Φ U . It is just a random linear combination Φ U ( x ) = X j ξ j ϕ j ( x ) , 6 H. HEDENMALM, P . MOZOL Y AKO, AND D. P ANOV where the set { ϕ j } forms an orthonormal basis of ℓ 2 ( U ), while { ξ j } are indep endent standard Gaussian random v ariables (i.e. distributed as N (0 , 1)). Since there are only finitely many vertices in U , w e see that Φ U can b e though t of as a random Gaussian v ector (Φ U ( x 1 ) , ...., Φ U ( x M )), where U = { x 1 , ..., x M } . W e define the discrete GFF 0 on the subgraph Γ U , denoted Ψ U , in a similar fashion: Ψ U ( x ) = X j ξ j ψ j ( x ) , x ∈ U, where the finite sequence { ψ j } j forms an orthonormal basis of H 1 0 ( U ), and { ξ j } j are independent standard Gaussian random v ariables. As in the case of the discrete WNF, the discrete GFF 0 can also b e viewed as a random Gaussian v ector. The follo wing prop osition sho ws that the distributions of Φ U and Ψ U do not depend on the c hoice of the orthonormal bases, since jointly Gaussian vectors are characterized b y first and second momen ts (including cov ariances). W e recall the standard notation co v( X , Y ) = E ( X Y ) − E ( X ) E ( Y ) for the co v ariance of tw o sto chastic v ariables, where E is the exp ectation op eration. Prop osition 3.1. F or any x, y ∈ U we have: ( i ) E (Φ U ( x )) = E (Ψ U ( x )) = 0 , ( ii ) co v(Φ U ( x ) , Φ U ( y )) = δ y ( x ) , and ( iii ) co v(Ψ U ( x ) , Ψ U ( y )) = 1 π ( y ) G U ( x, y ) . Pr o of. Since the Gaussian v ariables ξ j are from N (0 , 1), the first momen ts v anish, i.e. ( i ) holds. As to second momen ts, we find using the indep endence of the Gaussian v ariables that E Φ U ( x )Φ U ( y ) = X j ϕ j ( x ) ϕ j ( y ) = δ y ( x ) , where the last equality identit y uses the orthonormal basis prop erty of { ϕ j } j in ℓ 2 ( U ). In a similar fashion, we find that E Ψ U ( x )Ψ U ( y ) = X j ψ j ( x ) ψ j ( y ) = 1 π ( y ) G U ( x, y ) , where the last equality uses the orthonormal basis prop ert y of { ψ j } j in the discrete Sob olev space H 1 0 ( U ). □ 3.3. Desirable properties of the sought-after op erator. Given Γ U as ab ov e, we consider a connected subgraph Γ W of Γ U with v ertex set W . F or simplicit y w e assume that Γ U ’prop erly en v elops’ Γ W : W and Γ \ U are disconnected in Γ, but for every x ∈ U \ W there exists an edge ( x, y ) in Γ U for some y ∈ W (so that U \ W forms a ’la y er’ around W ). The orthonormal basis in ℓ 2 ( U ) is tak en to b e just the collection of p oint masses in vertices of U , { ϕ n } := { δ x } x ∈ U . W e aim to define an op erator Q U,W = Q W acting on ℓ 2 ( U ) suc h that Q W Φ U = Ψ W holds in the sense of having equal probabilit y distributions. No w, there happen to b e a whole lot of suc h possible op erators, so w e wish to narrow our search by requiring it to satisfy a certain set of additional conditions: ( a ) Isometry . The operator Q W should map the basis { δ x } x ∈ W to some orthonormal basis { ψ x } x ∈ W of H 1 0 ( W ). ( b ) Boundary v alues. Since Ψ W v anishes outside of W , so that supp Q W f ⊂ W , it is conv enien t to assume that Q W actually do es not take v alues of f outside W into accoun t, so that Q W f ≡ 0 , if supp f ⊂ U \ W. DGFF VIA HADAMARD 7 ( c ) Marko v prop erty . Let W ′ ⊂ W b e ’en v elop ed’ b y W in the abov e sense. The conditional exp ectation φ W,W ′ := E (Ψ W | Ψ W 1 W \ W ′ ) , whic h is random b ecause it is conditioned on random v alues in W \ W ′ , is almost surely discrete harmonic on W ′ . Its b oundary v alues are those of Ψ W on W \ W ′ , and we also hav e that Ψ W − φ W,W ′ = Ψ W ′ holds in distribution. This suggests that w e should assume Q W f − Q W ′ f to b e discrete harmonic on W ′ with b oundary v alues Q W f . Moreo v er, by the previous condition, if f is supp orted on W \ W ′ , then Q W f − Q W ′ f = Q W f , in particular Q W δ x is discrete harmonic on W ′ , x ∈ W \ W ′ . F or point masses in W ′ w e actually ask even more, namely stability , Q W δ x = Q W ′ δ x , x ∈ W ′ , so that the difference v anishes: Q W δ x − Q W ′ δ x = 0 (and in particular it is discrete harmonic). It turns out that there is natural w a y to construct suc h an operator – b y gro wing it from a giv en p oin t. Although the detailed construction is presen ted in the next section, w e supply a general outline below. W e assume that the op erator Q W ′ is given for some subset of vertices W ′ ⊂ W , and that it supplies an isometry ℓ 2 ( W ′ ) → H 1 0 ( W ′ ), i.e. Q W ′ δ x = ψ x , x ∈ W ′ , for some choice of the orthonormal basis { ψ x } x ∈ W ′ of H 1 0 ( W ′ ). As usual, we extend ψ x ( y ) := 0 for x ∈ W ′ and y ∈ W \ W ′ . Next, w e consider the collection of Kronec k er deltas { δ ξ } ξ ∈ W \ W ′ whic h together with the basis of ℓ 2 ( W ′ ) supplies an orthonormal basis for ℓ 2 ( W ). W e aim to define the action of Q W on these functions by putting Q W δ ξ = ψ ξ for ξ ∈ W \ W ′ , where the functions ψ ξ for ξ ∈ W \ W ′ are unit v ectors that are mutually orthogonal in H 1 0 ( W ) ⊖ H 1 0 ( W ′ ), where H 1 0 ( W ′ ) is – as usual – though t of as a subspace of H 1 0 ( W ) by extending the functions to v anish in W \ W ′ . The th us extended basis { ψ x } x ∈ W will naturally form an orthonormal basis of H 1 0 ( W ), whic h in turn allo ws us to go to the next step in the iteration pro cedure. The P oisson k ernel for W , ˜ ψ ξ ( y ) := P W,W \ W ′ ( y , ξ ) where ξ ∈ W \ W ′ , are linearly indep enden t, and moreov er, orthogonal to the subspace H 1 0 ( W ′ ) = span { ψ x : x ∈ W ′ } . Indeed, w e calculate that ⟨ ˜ ψ ξ , ψ x ⟩ ∇ ,W = ⟨ d ∗ d ˜ ψ ξ , ψ x ⟩ W = 0 , ξ ∈ W \ W ′ , x ∈ W ′ , since, as a matter of definition, d ∗ d ˜ ψ ξ = 0 holds on supp ψ x ⊂ W ′ . While the functions ˜ ψ ξ , ξ ∈ W \ W ′ are not, generally sp eaking, orthogonal to each other, one can easily mak e them orthogonal, via an orthogonalization pro cedure suc h as Gram-Schmidt. This means that there exists a w eight function ρ : W \ W ′ × W \ W ′ → R such that the functions ψ ξ ( y ) := X η ˜ ψ η ( y ) ρ ( η , ξ ) , ξ ∈ W \ W ′ , y ∈ W , form an orthonormal system in H 1 0 ( W ), and, combined with the system { ψ x } x ∈ W ′ , form an orthonor- mal basis for the space H 1 0 ( W ). Alternativ ely , if w e mak e the minimal iteration step of adding a single p oin t to W \ W ′ at a time, no additional orthogonality condition is required, and we would just normalize ˜ ψ ξ : ψ ξ = ∥ ˜ ψ ξ ∥ − 1 ∇ ,W ˜ ψ ξ for ξ ∈ W \ W ′ . 4. Discrete Hadamard’s formula and the rela ted opera tor 4.1. V ariation of the discrete Green function. W e recall the notation of the discrete Green k ernel and Poisson k ernel G n = G V (Γ n ) , P n = P V (Γ n ) ,V ( γ n ) , 8 H. HEDENMALM, P . MOZOL Y AKO, AND D. P ANOV in the context of a discrete foliation { γ n } n of the given graph Γ, where { Γ n } n denotes the asso ciated gro wth clusters. Lemma 4.1. (V ariation of the Green function) F or x, y ∈ V we have that G n ( x, y ) − G n − 1 ( x, y ) = X ξ ∈ V ( γ n ) P n ( x, ξ ) G n ( ξ , y ) . Pr o of. If one of x and y is off V (Γ n ), or b oth, then b oth sides of the iden tit y v anish. If not, then b oth are in V (Γ n ), and w e think of y as fixed. W e observ e that d ∗ d  G n ( · , y ) − G n − 1 ( · , y )  = 0 on V (Γ n − 1 ) = V (Γ n ) \ V ( γ n ) , while G n ( x, y ) − G n − 1 ( x, y ) = G n ( x, y ) , x ∈ V ( γ n ) , so the claimed identit y now follo ws from the standard discrete P oisson representation form ula. □ 4.2. Hadamard v ariation form ula in the discrete setting. W e no w pro ceed to deriv e the dis- crete version of the Hadamard’s v ariational formula. T o prepare the ground for that, we define the b oundary normalized Green op erator ˜ G ⟨ n ⟩ : ℓ 2 ( V ( γ n )) → ℓ 2 ( V ( γ n )) by (4.1) ˜ G ⟨ n ⟩ f ( x ) = X y ∈ V ( γ n ) ˜ G n ( x, y ) f ( y ) = X y ∈ V ( γ n ) G n ( x, y ) f ( y ) π ( y ) , x ∈ V ( γ n ) . Since ˜ G n ( x, y ) is symmetric and defines a positive op erator ˜ G n on ℓ 2 ( V (Γ n )), the b oundary op erator ˜ G ⟨ n ⟩ is definitely symmetric and semi-p ositive. Later on, we will need to know that it is actually p ositiv e, and hence inv ertible. Prop osition 4.1. The b oundary Gr e en op er ator ˜ G ⟨ n ⟩ : ℓ 2 ( V ( γ n )) → ℓ 2 ( V ( γ n )) given by (4.1) is symmetric with p ositive eigenvalues. In p articular, ˜ G ⟨ n ⟩ is invertible. Pr o of. Given that the op erator is symmetric and semi-p ositive, w e just need to exclude 0 as an eigen v alue. In other w ords, we show that for f ∈ ℓ 2 ( V ( γ n )), we hav e the implication ˜ G ⟨ n ⟩ f = 0 = ⇒ f = 0 . W e extend f ∈ ℓ 2 ( V γ n )) to the growth cluster V (Γ n ) b y declaring that f = 0 on V (Γ n ) \ V ( γ n ) = V (Γ n − 1 ), and consider the function ˜ G n f , whic h is in ℓ 2 ( V (Γ n )). Since as a matter of definition, ˜ G n f = ˜ G ⟨ n ⟩ f on V ( γ n ), w e know that ˜ G n f = 0 on V ( γ n ). But we also know that ˜ G n f is harmonic in V (Γ n ) \ supp f ⊃ V (Γ n − 1 ). By the uniqueness of the P oisson extension then, we obtain that ˜ G n f = P n ( G ⟨ n ⟩ f ) = 0 on V (Γ n ) , and hence f = ∆ V (Γ n ) ˜ G n f = 0 . The pro of is complete. □ No w, according to the sp ectral theorem for real symmetric matrices there is an orthonormal basis of real eigen v ectors. If λ ∈ R > 0 , and f λ ∈ ℓ 2 ( V ( γ n )) is a non trivial real-v alued eigenv ector, so that ˜ G ( n ) f λ = λf λ holds, w e can simply declare the canonical square root op erator ˜ R n : ℓ 2 ( V ( γ n )) → ℓ 2 ( V ( γ n )) by the formula ˜ R n f λ = √ λf λ , where we c hoose the nonnegativ e square ro ot of λ . By linear extension, this giv es a well-defined op erator, with the prop erty that ( ˜ R n ) 2 = ˜ G ( n ) . It is also necessarily real and symmetric: If w e put ˜ R n ( x, y ) := ˜ R n δ x ( y ) , x, y ∈ V ( γ n ) , DGFF VIA HADAMARD 9 then ˜ R n ( x, y ) ∈ R , and ˜ R n ( x, y ) = ˜ R n ( y , x ) , x, y ∈ V ( γ n ) . The prop erty of b eing the op erator square ro ot now can expressed in the follo wing form, for x, y ∈ V ( γ n ): (4.2) G n ( x, y ) π ( y ) = ˜ G n ( x, y ) = ˜ G ⟨ n ⟩ ( δ x )( y ) = ( ˜ R n ) 2 ( δ x )( y ) = X ξ ∈ V ( γ n ) ˜ R n ( x, ξ ) ˜ R n ( ξ , y ) . W e are now ready to form ulate the general discrete Hadamard v ariational form ula. Theorem 4.2. (Discrete Hadamard v ariational form ula) F or x, y ∈ V , we have ˜ G n ( x, y ) = G n ( x, y ) π ( y ) = X ξ ∈ V (Γ n ) K t ( ξ ) ( x, ξ ) K t ( ξ ) ( y , ξ ) , wher e t ( ξ ) denotes the unique discr ete time step m ∈ Z ≥ 0 such that ξ ∈ V ( γ m ) , and the indic ate d kernel is given by K m ( y , ξ ) := X η ∈ V ( γ m ) P m ( y , η ) ˜ R m ( η , ξ ) , ξ ∈ V ( γ m ) . Pr o of. In view of Lemma 4.1 and the discrete harmonicity of the Green function off the prescrib ed p ole, combined with the symmetry (2.2) and the identit y (4.2), we ha v e that G n ( x, y ) − G n − 1 ( x, y ) = X ξ ∈ V ( γ n ) P n ( x, ξ ) G n ( ξ , y ) = X ξ ∈ V ( γ n ) P n ( x, ξ ) G n ( y , ξ ) π ( y ) π ( ξ ) = X ξ ∈ V ( γ n ) P n ( x, ξ ) X η ∈ V ( γ n ) P n ( y , η ) G n ( η , ξ ) π ( y ) π ( ξ ) = π ( y ) X ξ ∈ V ( γ n ) P n ( x, ξ ) X η ∈ V ( γ n ) P n ( y , η ) X σ ∈ V ( γ n ) ˜ R n ( η , σ ) ˜ R n ( σ, ξ ) = π ( y ) X σ ∈ V ( γ n ) X ξ ∈ V ( γ n ) P n ( x, ξ ) ˜ R n ( σ, ξ ) X η ∈ V ( γ n ) P n ( y , η ) ˜ R n ( η , σ ) = π ( y ) X σ ∈ V ( γ n ) K n ( x, σ ) K n ( y , σ ) . Next, by summing b oth sides ov er n , w e obtain G n ( x, y ) − G 0 ( x, y ) = π ( y ) n X m =1 X σ ∈ V ( γ m ) K m ( x, σ ) K m ( y , σ ) . Note that for σ ∈ V ( γ 0 ) we hav e that K 0 ( x, σ ) = X η ∈ V ( γ 0 ) P 0 ( η , x ) ˜ R 0 ( η , σ ) = ˜ R 0 ( x, σ ) , and, consequently , it follo ws that π ( y ) X σ ∈ V ( γ 0 ) K 0 ( x, σ ) K 0 ( y , σ ) = π ( y ) X σ ∈ V ( γ 0 ) ˜ R 0 ( x, σ ) ˜ R 0 ( y , σ ) = π ( y ) ˜ G 0 ( x, y ) = G 0 ( x, y ) . In combination with the already obtained identit y , w e arrive at the decomp osition G n ( x, y ) = π ( y ) n X m =0 X σ ∈ V ( γ m ) K m ( x, σ ) K m ( y , σ ) . 10 H. HEDENMALM, P . MOZOL Y AKO, AND D. P ANOV Since the v ertex set V (Γ n ) of the foliation cluster is the disjoin t union of the v ertex sets of the foliating la y ers V ( γ m ), m = 0 , . . . , n , this formula is identical with that of the assertion of the theorem. □ 4.3. Definition of the Hadamard op erator. W e no w introduce the Hadamar d op er ator Q n : ℓ 2 ( V (Γ n )) → ℓ 2 ( V (Γ n )) as giv en by the form ula Q n f ( x ) := X y ∈ V (Γ n ) K t ( y ) ( x, y ) f ( y ) . Its adjoint with resp ect to the standard ℓ 2 ( V (Γ n ))-dualit y is then Q ∗ n f ( x ) = X y ∈ V (Γ n ) K t ( x ) ( y , x ) f ( y ) , x ∈ V (Γ n ) . W e note that for a ∈ V (Γ n ), a calculation gives that Q n Q ∗ n δ a ( z ) = X ξ ∈ V (Γ n ) K t ( ξ ) ( z , ξ ) K t ( ξ ) ( a, ξ ) = G n ( z , a ) π ( a ) = ˜ G n ( z , a ) = ˜ G n δ a ( z ) , a ∈ V (Γ n ) , where the middle iden tity relies on the discrete Hadamard v ariation formula of Theorem 4.2. Since a ∈ V (Γ n ) is arbitrary , it is no w immediate that Q n Q ∗ n = ˜ G n . This prop erty is a basic ingredient in the pro of of the follo wing theorem. Theorem 4.3. The Hadamar d op er ator Q n is an isometric isomorphism ℓ 2 ( V (Γ n )) → H 1 0 ( V (Γ n )) . Pr o of. W e recall the notational con ven tion ∆ n := ∆ V (Γ n ) , and observe that there is a well-defined canonical square ro ot ( ∆ n ) 1 2 whic h acts as a symmetric matrix with real en tries on the finite- dimensional space ℓ 2 ( V (Γ n )). F or f , g ∈ ℓ 2 ( V (Γ n )), we chec k that (4.3)  Q ∗ n ( ∆ n ) 1 2 f , Q ∗ n ( ∆ n ) 1 2 g  V (Γ n ) =  Q n Q ∗ n ( ∆ n ) 1 2 f , ( ∆ n ) 1 2 g  V (Γ n ) =  ˜ G n ( ∆ n ) 1 2 f , ( ∆ n ) 1 2 g  V (Γ n ) =  ( ∆ n ) 1 2 ˜ G n ( ∆ n ) 1 2 f , g  V (Γ n ) = ⟨ f , g ⟩ V (Γ n ) , since ˜ G n = ∆ − 1 n holds. This en tails that the op erator Q ∗ n ( ∆ n ) 1 2 : ℓ 2 ( V (Γ n )) → ℓ 2 ( V (Γ n )) is an isometry , which immediately entails that the adjoin t op erator ( ∆ n ) 1 2 Q n : ℓ 2 ( V (Γ n )) → ℓ 2 ( V (Γ n )) is a partial isometry , and, in particular, a contraction. Suppose we can show that the k ernel (or null space) of ( ∆ n ) 1 2 Q n is trivial. Then the op erator ( ∆ n ) 1 2 Q n m ust b e an isometry , in which case ⟨ Q n f , Q n g ⟩ ∇ , Γ n =  ( ∆ n ) 1 2 Q n f , ( ∆ n ) 1 2 Q n g  Γ n = ⟨ f , g ⟩ Γ n . In other words, w e would ha v e shown that the op erator Q n : ℓ 2 (Γ n ) → H 1 0 (Γ n ) defines an isometric isomorphism, as claimed. W e assume that f 0 ∈ ℓ 2 ( V (Γ n )) with ( ∆ n ) 1 2 Q n f 0 = 0 , and intend to show that f 0 = 0. Next, since ∆ n is inv ertible on ℓ 2 ( V (Γ n )), we kno w that ( ∆ n ) 1 2 is in v ertible too, and, consequen tly , that Q n f 0 = 0. F rom the definition of the Hadamard operator, we see that Q n f 0 ( x ) = X y ∈ V (Γ n ) K t ( y ) ( x, y ) f 0 ( y ) = X y ∈ V ( γ n ) K n ( x, y ) f 0 ( y ) , x ∈ V ( γ n ) , since K m ( x, y ) = 0 if x ∈ V ( γ n ) and y ∈ V ( γ m ) with m < n . Moreo ver, K n ( x, y ) = X η ∈ V ( γ n ) P n ( x, η ) ˜ R n ( η , y ) = ˜ R n ( x, y ) , x, y ∈ V ( γ n ) , DGFF VIA HADAMARD 11 since the Poisson k ernel at the discrete boundary is rather trivial: P n ( x, η ) = δ x ( η ) for x, η ∈ V ( γ n ). As a consequence, we hav e that Q n f 0 ( x ) = X y ∈ V ( γ n ) ˜ R n ( x, y ) f 0 ( y ) = ˜ R n [ f ⟨ n ⟩ 0 ]( x ) , x ∈ V ( γ n ) , where w e use the notation f ⟨ n ⟩ 0 to denote the restriction of f 0 to the foliation la yer V ( γ n ). F rom our assumption w e know that Q n f 0 = 0, and hence it follows that ˜ R n [ f ⟨ n ⟩ 0 ] = 0. Next, since ˜ R n is inv ertible in view of Prop osition 4.1 and the definition of ˜ R n as the canonical self-adjoint square ro ot of ˜ G ⟨ n ⟩ , it is immediate that f ⟨ n ⟩ 0 = 0, whic h means that supp f 0 ⊂ V (Γ n − 1 ). But then we may observ e that Q n f 0 ( x ) = X y ∈ V (Γ n ) K t ( y ) ( x, y ) f 0 ( y ) = X y ∈ V ( γ n − 1 ) K n − 1 ( x, y ) f ( y ) , x ∈ V ( γ n − 1 ) , whic h puts us in the analogous situation in the preceding foliation lay er γ n − 1 . If w e pro ceed w e the same argumen ts as for the la y er γ n , w e obtain that supp f 0 ⊂ V (Γ n − 2 ). By iteration n steps, w e obtain that supp f 0 ⊂ V ( γ 0 ), and if w e rep eat the same argument once more, we find that ˜ R n f ⟨ 0 ⟩ 0 ] = 0 which in turn giv es f ⟨ 0 ⟩ 0 = 0 and hence that f 0 = 0 iden tically . The pro of of the theorem is now complete. □ 5. Gro wing the Discrete Gaussian Free Field using white noise as building blocks 5.1. F orm ulation of the m ain result. In this Section w e formulate and pro v e our main result, whic h is analogous to [HN, Theorem 5.1]. In terms of notation, we write Q n f = Q n [ 1 V (Γ n ) f ] for a function f : V → R not necessarily supp orted on V (Γ n ). Theorem 5.1. L et Φ denote the White Noise Field on the gr aph Γ N , wher e N = 0 , 1 , 2 , . . . is given. F or n = 0 , 1 , . . . , N , we define the sto chastic fields Ψ n = Q n Φ . Then Ψ n is the Gaussian F r e e Field on Γ n . Mor e over, the incr ements of the pr o c ess Ψ n with r esp e ct to the p ar ameter n ar e indep endent. Mor e pr e cisely, if n i ∈ Z ≥ 0 ar e liste d incr e asingly with 0 ≤ n 0 < ... < n k ≤ N , then the k r andom ve ctors (Ψ n i ( x ) − Ψ n i − 1 ( x )) x ∈ V (Γ N ) for i = 1 , . . . , k ar e al l sto chastic al ly indep endent. Pr o of. Since the distribution of Φ do es not dep end on the orthonormal basis of ℓ 2 (Γ N ) that defines the field, w e may as w ell assume use the Kronec ker delta basis, so that Φ( x ) = X y ∈ V (Γ N ) ξ y δ y ( x ) , x ∈ V (Γ N ) , where, as usual, { ξ y } y ∈ V (Γ N ) are indep endent standard real Gaussian random v ariables. It then follo ws that Ψ n ( x ) = Q n Φ( x ) = X y ∈ V (Γ N ) ξ y Q n δ y ( x ) , x ∈ V (Γ N ) . According to Theorem 4.3, Q n : ℓ 2 ( V (Γ n )) → H 1 0 ( V (Γ n )) is an isometric isomorphism, and, conse- quen tly , the set { Q n δ y } y ∈ V (Γ n ) forms an orthonormal basis of H 1 0 (Γ n ). So, Ψ n is indeed the Gaussian F ree Field on Γ n . Next, as to the increments, w e hav e Ψ n i ( x ) − Ψ n i − 1 ( x ) = X y ∈ V (Γ n i ) \ V (Γ n i − 1 ) ξ y Q n i δ y ( x ) , as it is clear from the definition of Q n that for 0 ≤ k 1 < k 2 and f ∈ ℓ 2 ( V (Γ k 1 )) ⊂ ℓ 2 ( V (Γ k 2 )), we hav e Q k 1 f = Q k 2 f . The required indep endence follows from the sto chastic indep endence of the individual Gaussians in the family { ξ y } y . □ 12 H. HEDENMALM, P . MOZOL Y AKO, AND D. P ANOV 6. Proper ties of the constructed Gaussian free field Ψ n 6.1. Harmonically extended weigh ted white noise as time-step incremen ts. In this Section w e discuss some interesting prop erties of the Gaussian F ree Field constructed in Theorem 5.1. W e first consider the discrete time-deriv ativ e of Ψ n , the Gaussian F ree Field on the foliation cluster Γ n pro duced from the White Noise Φ on the bigger graph Γ N , 0 < n ≤ N . F or any x ∈ V , w e ha ve that Ψ n ( x ) − Ψ n − 1 ( x ) = Q n Φ( x ) − Q n − 1 Φ( x ) = X y ∈ V ( γ n ) K n ( x, y )Φ( y ) = X y ∈ V ( γ n ) X η ∈ V ( γ n ) P n ( x, η ) ˜ R n ( η , y )Φ( y ) = X η ∈ V ( γ n ) P n ( x, η ) X y ∈ V ( γ n ) ˜ R n ( η , y )Φ( y ) . While the left-hand side expresses the discrete time deriv ative of Ψ n , the right-hand side can b e understo od as the P oisson extension of a weigh ted White Noise Field on the foliation la y er γ n . Note that this result finds its t w o-dimensional smooth contin uous analogue in [HN]. Also, it is not hard to see that a similar computation for the difference Ψ n − Ψ m with 0 ≤ m < n ≤ N gives the well known Mark o v property of the Gaussian F ree Field. 6.2. The connection with Brownian motion. W e turn to another prop ert y of the constructed GFF 0 . First, we fix a vector f ∈ ℓ 2 (Γ N ). Then the random v ector ( ⟨ f , Ψ n ⟩ Γ n ) n where n = 0 , . . . , N has a Gaussian distribution (meaning that the entries are join tly Gaussian). W e will compute its distribution explicitly , moreov er we show that it is the same as that of a w eighted one-dimensional random walk (cf. the corresp onding result in [HN]). Prop osition 6.1. F or f ∈ ℓ 2 ( V (Γ N )) , we have that  ⟨ f , Ψ n ⟩ V (Γ n )  n d =  B  ∥ Q ∗ n f ∥ 2 V (Γ n )   n d =  n X k =0  B ( k + 1) − B ( k )  ∥ 1 V ( γ k ) Q ∗ n f ∥ V (Γ n )  n wher e n r anges over { 0 , . . . , N } and ” d = ” me ans e quality in pr ob ability distribution, while B ( · ) stands for a standar d c ontinuous Br ownian motion on R ≥ 0 with initial value B (0) = 0 . Pr o of. F or brevity of notation, we write F n ( f ) := ⟨ f , Ψ n ⟩ V (Γ n ) = X y ∈ V (Γ n ) ξ y f ( y ) , where we recall that the ξ y are all indep enden t standard Gaussian random v ariables. W e find that E F n ( f ) = X y ∈ V (Γ n ) f ( y ) E ξ y = 0 and F n ( f ) = ⟨ f , Ψ n ⟩ V (Γ n ) = ⟨ f , Q n Φ ⟩ V (Γ n ) = ⟨ Q ∗ n f , Φ ⟩ V (Γ N ) . In addition to the first moments, we shall need the second momen ts as well. F or n ≤ m ≤ N , we calculate that E ( F n ( f ) F m ( f )) = E  ⟨ Q ∗ n f , Φ ⟩ V (Γ N ) ⟨ Q ∗ m f , Φ ⟩ V (Γ N )  = ⟨ Q ∗ n f , Q ∗ m f ⟩ V (Γ N ) = X x ∈ V (Γ N ) Q ∗ n f ( x ) Q ∗ m f ( x ) . W e note that Q ∗ n f = 0 holds on V (Γ N ) \ V (Γ n ), while Q ∗ n f = Q ∗ m f holds on V (Γ n ). so that by the ab o v e calculation, it follows that E ( F n ( f ) F m ( f )) = ⟨ Q ∗ n f , Q ∗ n f ⟩ Γ N = ∥ Q ∗ n f ∥ 2 V (Γ N ) = ∥ Q ∗ n f ∥ 2 V (Γ n ) , n ≤ m ≤ N . DGFF VIA HADAMARD 13 The first and second momen t structure of a standard Bro wnian motion B ( t ) with B (0) = 0 is E B ( t ) = 0 and E ( B ( t ) B ( t ′ )) = min { t, t ′ } for t, t ′ ∈ R ≥ 0 . This means that  B ( ∥ Q ∗ n f ∥ 2 V (Γ n ) )  n has the same first and second momen ts (including co v ariances) as ( F n ( f )) n , so that by (joint) normality , the first equality in distribution follows. Next, w e put W n ( f ) := n X k =0  B ( k + 1) − B ( k )  ∥ 1 V ( γ k ) Q ∗ n f   V (Γ n ) . Then E W n ( f ) = 0, and regarding the second moments, we find that for n ≤ m ≤ N , (6.1) E ( W n ( f ) W m ( f )) = n X k =0   1 V ( γ k ) Q ∗ n f   V (Γ n ) ∥ 1 V ( γ k ) Q ∗ m f   V (Γ m ) = n X k =0   1 V ( γ k ) Q ∗ n f   2 V (Γ n ) =   Q ∗ n f   2 V (Γ n ) . The second equalit y in distribution no w follows as well. □ W e recall the P oisson extension op erator P n : ℓ 2 ( γ n ) → ℓ 2 (Γ n ) given by P n f ( x ) = X y ∈ V ( γ n ) P n ( x, y ) f ( y ) , x ∈ Γ n , where P n = P V (Γ n ) ,V ( γ n ) , and denote by P ∗ n its adjoin t op erator ℓ 2 (Γ n ) → ℓ 2 ( γ n ). It is is sometimes called the swe ep op er ator , at least in the con tin uous context, and it is giv en by the formula P ∗ n f ( y ) = X x ∈ V (Γ n ) P n ( x, y ) f ( x ) , y ∈ γ n . W e are no w ready to state a result concerning the b eha vior of the b oundary a verages of the GFF 0 pro cess, analogous to what w as obtained in the t wo-dimensional con tin uous smo oth setting in [HN]. Prop osition 6.2. Supp ose n 1 , n 2 ∈ { 1 , . . . , N } and that n 1 < n 2 . Assume, mor e over, that f ∈ ℓ 2 (Γ n 1 ) ⊂ ℓ 2 (Γ N ) . We then have that  ⟨ P ∗ n f , Ψ n 2 ⟩ ℓ 2 ( γ n )  n d =  B    1 V (Γ n 2 ) \ V (Γ n ) Q ∗ n 2 f   2 V (Γ n 2 )  n d = n 2 X k = n  B ( k + 1) − B ( k )    1 V ( γ k ) Q ∗ n 2 f   V (Γ n 2 )  ! n , wher e n r anges over over n = n 1 , . . . , n 2 . Her e, B ( · ) denotes a standar d c ontinuous Br ownian motion on R ≥ 0 starting at B (0) = 0 . Pr o of. W e write A n ( f ) := ⟨ P ∗ n f , Ψ n 2 ⟩ V ( γ n ) = ⟨ f , P n Ψ n 2 ⟩ V (Γ n ) , where it is understo o d that P n Ψ n 2 v anishes on V (Γ N ) \ V (Γ n ). W e iden tify P n Ψ n 2 with Ψ n 2 − Ψ n − 1 on V (Γ n ), simply because it is harmonic on V (Γ n − 1 ) and has the same v alue on the b oundary la y er V ( γ n ). It no w follows that A n ( f ) =  f , 1 V (Γ n 1 ) (Ψ n 2 − Ψ n − 1 )  V (Γ n 1 ) = ⟨ f , Ψ n 2 ⟩ V (Γ n 2 ) − ⟨ f , Ψ n − 1 ⟩ V (Γ n − 1 ) = F n 2 ( f ) − F n − 1 ( f ) , where the notation F k ( f ) is an in the pro of of Proposition 6.1. The first momen ts v anish: E ( A n ( f )) = 0. W e pro ceed to consider the second moments. The second moments of F k ( f ) are known from the 14 H. HEDENMALM, P . MOZOL Y AKO, AND D. P ANOV pro of of Prop osition 6.1, and we can use them to obtain the second moments of A n ( f ). The result is that if n ≤ m , E ( A n ( f ) A m ( f )) = ∥ Q ∗ n 2 ∥ 2 V (Γ n 2 ) − ∥ Q ∗ m − 1 ∥ 2 V (Γ m − 1 ) . W e these first and second moments with the momen ts whic h are obtained for the other tw o pro cesses in v olving Bro wnian motion. The details are analogous with the calculations in the pro of of Prop osition 6.1 and for this reason omitted. □ References [LP] Ly ons, R., Peres, Y., Pr ob ability on tr e es and networks , Cambridge Series in Statistical and Probabilistic Mathematics, 42 . Cam bridge Universit y Press, New Y ork, 2016. [DLP] Ding, J., Lee, J. R., Peres, Y., Cover times, blanket times, and majorizing me asur es , In Pro ceedings of the fort y-third annual ACM symp osium on Theory of computing, pp. 61-70. ACM, New Y ork, 2011. [HN] Hedenmalm, H., Nieminen, P ., The Gaussian fre e field and Hadamar d’s variational formula , Probab. Theory Related Fields, 159 (2014), pp. 61-73. [B] Biskup, M., Extr ema of the two-dimensional Discr ete Gaussian F r e e Field , In: M. Barlo w and G. Slade (eds.): Random Graphs, Phase T ransitions, and the Gaussian F ree Field. SSPROB 2017. Springer Pro ceedings in Mathematics & Statistics, vol 304 , pp 163-407. Springer, Cham (2020). (H. Hedenmalm) Dep ar tment of Ma thema tics and Computer Science, St. Petersburg University, R us- sia, 14 line of the VO, 29B, 199178 St. Petersburg, R ussia; Beijing Institute of Ma thema tical Sciences and Applica tions, 101408, Hef angkou Village, Huaibei Town, Huairou District, 544, Beijing, China; De- p ar tment of Ma thema tics and St a tistics, University of Reading, U. K. Email address : haakan00@gmail.com (P . Mozolyak o) Dep ar tment of Ma thema tics and Computer Science, St. Petersburg University, R ussia, 14 line of the VO, 29B, 199178 St. Petersburg, Russia; Beijing Institute of Ma thema tical Sciences and Applica tions, 101408, Hef angkou Village, Huaibei Town, Huair ou District, 544, Beijing, China. Email address : pmzlcroak@gmail.com (D. Pano v) Dep ar tment of Ma thema tics and Computer Science, St. Petersbur g University, R ussia, 14 line of the VO, 29B, 199178 St. Petersburg, R ussia Email address : panovdan2003@gmail.com