1 Introduction

# Asymptotic Determinant of Discrete Laplace-Beltrami Operators

## Abstract.

We study combinatorial Laplacians on rectangular subgraphs of that approximate Laplace-Beltrami operators of Riemannian metrics as . These laplacians arise as follows: we define the notion of a Riemmanian metric structure on a graph. We then define combinatorial free field theories and describe how these can be regarded as finite dimensional approximations of scalar field theory. We focus on the Gaussian field theory on rectangular subgraphs of and study its partition function by computing the asymptotic determinant of the discrete laplacian.

## 1. Introduction

On a graph , the graph laplacian is an operator that acts on the space of functions as:

 Δf(u)=∑[u,v]∈Ef(v)−f(u) (1.1)

The determinants of graph laplacians have been well studied in many different contexts. For instance, in spectral graph theory, Kirchoff’s theorem relates the determinant of the graph laplacian on to the number of rooted spanning trees on . Similarly, by Temperley’s trick and its generalizations, the determinant of the graph laplacian on certain graphs computes the number of perfect matchings on a related graph [KPW00]. In each case, the determinant of the laplacian is the partition function of an associated model in statistical mechanics.

When is the graph of a regular lattice, the asymptotics of the determinant have been investigated in many papers. In particular, when is a rectangular subgraph of with vertices, the determinant of the laplacian (with Dirichlet boundary conditions) as is computed by [DD] in the context of Hamiltonian walks on the grid. The asymptotic expansion is:

 logdetΔNM=NM4Gπ+2(N+M)log(√2−1)+14log(NM)+O(1) (1.2)

where is the Catalan constant. More generally, the asymptotics of the determinants of laplacians on rectilinear regions of have been computed in [Ken00]. When appropriately rescaled, this graph laplacian can be understood as a finite difference approximation of the laplacian of Eulidean space.

In this paper, we study a generalized graph laplacians in that can be regarded as finite difference approximations of Laplace-Beltrami operators induced by a Riemannian metric . These laplacians arise as follows: on any graph , the cochain complex can be regarded as a combinatorial version of the de Rham complex. We define a Riemannian structure on as a set of inner products on the cochain spaces. On such a graph with Riemannian structure, we define a combinatorial Guassian field theories, as outlined in section §2.3. The partition function of the boson field theory is , where the combinatorial laplacian is by for all .

Assume now that the underlying space of is endowed with a metric . By square subdivision of , we have a sequence of graphs with mesh . Assume that the have Riemmanian structure such that the combinatorial laplacian on is a good approximation of the laplacian induced by on (see section 2.2.3). This can be regarded as discretization of a massless scalar field theory on a nontrivial background metric. We show that the determinant of has the asymptotic expansion (Theorem 3.1):

 logdetΔi=1ϵ2i∫MF(g)dxdy+1ϵi∫∂MB(g)⋅ds+12log(ϵi)+O(1) (1.3)

where the functions and are given in equation (3.1).

The discrete Guassian field theories are known to be conformally invariant in the scaling limit , and consequently the asymptotic expansion of the form (1.3) is generally anticipated by the theory of finite size scaling [CaPes]. In particular, the coefficient of the logarithmic term is universally related to the universal charge of the limiting conformal field theory and the Euler characteristic of the .

Some work has been done on relating the constant terms in the asymptotics of the determinant of discretized or finite difference laplacians with regularized functional determinants of analytic laplacians. In one-dimension, there are many positive results, for example [For92], [BFK]. In two dimensions, for some particular cases such as flat rectangles and flat torii, with particular choice of discretized laplacians, the constant term is known to be equal to the zeta-regularized determinant of the analytic laplacian, [DD],[CJK10]. We do not attempt to calculate the constant term in the asymptotic determinant (1.3) and its investigation for further work.

The structure of the paper is as follows. In §2, we describe the notion of a Riemannian structure on a graph, and define the combinatorial codifferential and combinatorial laplacian. We explain how a Riemannian manifold can be approximated by a sequence of graphs with Riemannian structure. Then we outline the definition of a combinatorial Gaussian free field theory on a graph and describe the relation to the dimer model of statistical physics. In §3, we focus on the bosonic field theory on , when is a rectangular region in endowed with a diagonal metric , and set up the calculation of the determinant. Sections §4 and §5 are more technical sections: §4 is devoted to estimating the inverse as ; in §5 we prove the main theorem, equation (1.3). Finally in §6, we conclude with some general remarks and further research directions.

### 1.1. Acknowledgements

I would like to thank my adviser Nicolai Reshetikhin for suggesting to study asymptotics of determinants of laplacians and for his reading of several drafts. I also thank Niccolo S. Poulsen for many valuable discussions regarding combinatorial Laplacians and dimer models, and for proofreading some versions of this text.

## 2. Preliminaries

Let be a cell decomposition of a simply connected region , with -cells for . The space of -chains for is the real vector spaces with canonical basis . The dual space space is the space of functions . There is an isomorphism induced by the canonical basis on . For any subcomplex , there is a natural projection given by restriction to . Define . The boundary of is the 1-dimensional subcomplex contained in .

On an oriented complex, there is a natural boundary operator . The coboundary is the adjoint of with respect to the pairing . The coboundary operator satisfies and the complex can be regarded as a discrete analog of the de Rham complex of a smooth manifold.

In the remainder, we assume for simplicity that is embedded such that image of each -cell is a simple convex polygon. In this case the -skeleton of then defines a simple planar graph embedded in .

#### The dual and double of a graph

We define the dual complex as the complex found by truncating the standard dual, so that is also a cell complex of (see figure 1). Note that as usual, there is a bijection from and to , but in addition the truncation gives a bijection of and .

The double of is, roughly speaking, the union of the and (see figure 1). Precisely, is the complex consisting of: a -cell at the center of each ; a -cell for each pair with adjacent to ; and a -cell , for each set of , , pairwise adjacent. Note that the graph of has a natural bipartite structure given by painting white if is a -cell, and black otherwise.

### 2.1. Riemannian Structures on Graphs

Recall that the metric on a compact, oriented Riemannian manifold induces a scalar product on the de Rham complex : for

 ⟨f,g⟩Ωq(M)=∫Mf∧∗g

where is the Hodge dual. Continuing to view the cochain spaces as the combinatorial analog of , we define:

###### Definition 2.1.

A Riemannian structure on is a set of scalar products for .

The scalar products can be required to satisfy certain additional axioms. In particular, recall that the scalar product on is local in the sense that it is the integral of a fiberwise scalar product. We consider more generally the combinatorial notion of a fiberwise map as follows:

###### Definition 2.2.

A map is said to -local if the matrix elements in the canonical basis of vanishes whenever the graph theoretic distance between is greater than .

###### Definition 2.3.

A Riemannian structue is local if the corresponding dual operator is -local for some . A Riemannian structure is said to be diagonal if it is -local.

A diagonal Riemannian structure can be understood as an assignment of a volume weight to each cell .

#### Approximation theory

We now consider how a Riemannian manifold can be approximated by a graph with Riemannian structure. We assume as before that is a cell decomposition of a region , but in addition assume that is endowed with a smooth Riemannian metric . Define the mesh of is . The embedding of in defines a surjection, namely the de Rham integration map :

 IXω=∑α∈Xqα∫αω

for .

Let be a sequence of complexes in , such that , for example generated by iterated subdivision of an initial complex. A sequence of Riemannian structures on the is said to be consistent if the integration maps approaches an isometry of spaces and as . More precisely:

###### Definition 2.4.

A sequence of Riemannian structures on is consistent if for any there exists a constant such that, for all :

 ∣∣⟨IXif,IXif⟩Cq(Xi)−⟨f,f⟩Ωq(M)∣∣≤ηfϵXi

### 2.2. The Combinatorial Laplacian

Let be a complex with a Riemannian structure. Let be a subcomplex of . Let where is orthogonal projection. The combinatorial codifferential combinatorial codifferential is defined the adjoint of :

 ⟨dq,Lf,g⟩Cq+1(X,L)=⟨f,δq+1,Lg⟩Cq(X,L)

for all .

###### Remark 2.5.

Choosing the canonical basis for the and writing the scalar products as matrices , the codifferential can be written in matrix form as:

 δq=g−1q−1dTqgq

From this, it is clear that generally is -local (for any !) only if the scalar product is diagonal.

#### Duality

In this section, we briefly explain how Hodge duality arises in this combinatorial framework. Let be a complex with a diagonal Riemannian structure, and as above . Let be the complement of the image of under the bijection , and be the complex found by removing from the image of under both bijections and .

There is a bijection of and , which induces an isomorphism . Putting a Riemannian structure on via the isomorphism, becomes an isomorphism of chain complexes and ; ie. and .

#### The Combinatorial Laplacian

The combinatorial Laplacian is defined as . This Laplacian should be understood as corresponding to mixed boundary conditions, with relative (Dirichlet) boundary conditions on and absolute (Neumann) boundary conditions on . In particular, since we focus attention on -forms, we let:

 ΔL=δ1,Ld0,L

For a diagonal Riemannian structure, it follows by straightforward computation that for , the combinatorial laplacian can be written:

 ΔLf(p)=∑σ∈X1p∩σ≠0σ∩L=0|σ||p|(f(q)−f(p)) (2.1)

#### Consistency of the Combinatorial Laplacian

Fix a submanifold , and let be the Laplace-Beltrami operator acting on functions with Dirichlet boundary conditions on and Neumann on . Consider a sequence of graphs approximating as in Section 2.1.1, and such that for each there is a subcomplex that is a decomposition of . In this context:

###### Definition 2.6.

The sequence of combinatorial laplacians on is said to be consistent if for any there exists a constant such that, for all :

 ∣∣ΔiLIXif−IXiΔf∣∣≤ηfϵ2Xi (2.2)

### 2.3. Gaussian Field Theories on Graphs

In this section, we briefly outline the definition of discrete Guassian field theories on graphs with Riemannian struture.

#### Bosonic Free Field Theory

Let be a complex with a Riemannian structure. The (massless) free field action, or Dirichlet energy, is the quadratic form :

 S(f)=⟨df,df⟩C1(X)

for .

The space of fields of the discrete field theory is given by prescribing boundary conditions as follows: let be a nonempty subcomplex and let .The Bosonic scalar field theory is then defined by the partition function:

 Zb(X,L)=∫He−S(ψ)Dψ

where the measure on is induced by the scalar product on . The correlation functions are defined in the usual way as:

 ⟨ψ(p1)⋯ψ(pn)⟩=1Zb(X,L)∫Hψ(p1)⋯ψ(pn)e−S(ψ)Dψ

Using standard Gaussian integral formulae give:

 logZb(X,L)=−12logdet(ΔL)+12(dimH)log(2π)

Similarly, correlation functions can be expressed in terms of matrix elements of . For exampe, the two point function:

 ⟨ψ(p1)ψ(p2)⟩=(Δ−1L)p1,p2

#### Fermionic Free Field Theory

As before, let be a subcomplex of . Let and . The combinatorial Dirac operator is the restriction of to , and the adjoint is the restriction of to . This operator should be regarded as a combinatorial analog of the Hodge-Dirac operator.

The space of fields for the free fermionic field theory is and . The theory is defined by partition function given by the Grassmanian integral over :

 Missing or unrecognized delimiter for \left (2.3)

where the measure is induced by the scalar products on and . Correlation functions are once again defined by insertions into the path integral:

 ⟨ψ(p1)χ(p1)⋯ψ(pn)χ(pn)⟩ =1Zf(X,L)∫H+ ∫H−ψ(p1)χ(p1)⋯ψ(pn)χ(pn)exp(12⟨ψ,DLχ⟩)Dχ

We can explicitly calculate as follows. Choosing the standard basis for and for , one can write the measure as:

 DχDψ=∏i1√|χi|dχi∏j1√|ψj|dψi

Using the standard Grassman integral formulae, the partition function (2.4) can be evaluated as:

 Zf(X,L)=∏√|ψi|∏√|χj|det(DL) (2.4)

The fermionic field theory is closely related to the bosonic theory. First define the dual fermionic theory as:

 ˜Zf(X,L)=∫H+∫H−exp(12⟨DLψ,χ⟩)DχDψ (2.5)

Then it is straightforward to show that:

 Zf(X,L)˜Zf(X,L)=(Zb(˜XL,˜L)Zb(X,L))−1

#### The Dimer Model

The free fermionic theory defined above is closely related to the dimer model of statistical mechanics. First we briefly recall the definition of the dimer model. On a graph with positive edge weights , a perfect matching of is a subset of edges such that each vertex is contained in exactly one edge . The weight of the matching is . This defines a Gibbs measure on the set of all perfect matchings: the probability of a matching is:

 μ(M,w)=w(M)Z(G,w)

where the partition function . A gauge transformation of the dimer model is a function that acts on the edge weights as . The gauge transformation leaves the measure invariant, but transforms the partition function as:

 Z(G,(λ⋅w))=⎛⎝∏p∈G0λ(p)⎞⎠Z(G,w) (2.6)

The dimer model is exactly solvable by the Kasteleyn technique as follows: for simplicity, assume is bipartite with black vertices and white vertices . A Kasteleyn orientation is a function , such that the product of on edges around any face is either if the number of edges is divisible by four, or otherwise. The Kasteleyn matrix is the matrix with columns and rows indexed by black and white vertices respectively, and with matrix elements

 Kbiwj=˜w([biwj])w([bi,wj])

when and are adjacent, and zero otherwise.

Kasteleyn’s theorem is that the partition function of the dimer model is given by . In addition, the Kasteleyn matrix encodes the statistics of dimer model via the local statistics theorem, which gives correlation functions of the dimer model in terms of the inverse Kasteleyn matrix.

We now define a dimer model on the double of a complex endowed with a local Riemannian structure. As before, let be a non-empty subgraph of the boundary. Define an the edge weight function on the double as:

 wr([pσ,pτ]) = ⎷⟨σ,σ⟩Cq(G)⟨τ,τ⟩Cq+1(G)

In writing the above formula, we have assumed is of smaller dimension than in . Let be the graph found by attaching to each with a unit weighted edge and an additional univalent vertex. (See figure 2). Let be the dimer partition function of the dimer model defined by the graph with weight .

###### Proposition 2.7.
 Zd(X,wr,L)=Zf(X,L)

This proposition states essentially that is a Kasteleyn matrix for the dimer model . It follows then that the correlation functions of the fermionic theory also give the correlation functions of the dimer model.

###### Proof.

First, note that the dimer model on is equivalent to the dimer model found by removing from vertices with , and edges adjacent to them. In addition, applying the gauge transformation

 λ(pσ)={1/√|σ|if σ∈X0,X2√|σ|if σ∈X1

on the remaining vertices transforms the edge weights into:

 w′([pσ,pτ]) ={1 if σ∈X0,τ∈X1|σ|/|τ|if σ∈X1,τ∈X2

Comparing the action of the gauge transformation (2.6) and the explicit formula for the fermionic partition function (2.4), it remains to show that:

 Zd(D(X)∖L,w′)=det(DL)

Recall that the double has a bipartite structure. Regarding as map operator from white vertices in (edges in ) to black vertices (vertices and faces in we compute for :

 Dτ=∑σ∈X0(−1)(σ,τ)σ+∑σ∈X2(−1)(σ,τ)|σ||τ|σ

where is the function that is 0 if is not adjacent to , and is if the orientations of and are compatible. That is, up to a sign, the matrix element is equal to the edge weight . It remains to check that the signs of the matrix elements of satisfy the Kasteleyn orientation condition. Since every face in the double is quadrilateral, the Kasteleyn condition requires the number of minus signs around each is odd. This follows from and . ∎

## 3. Combinatorial Laplacians in Rectangular Graphs

In the remainder of this paper, we focus on the particular case that is a rectangular region with a diagonal metric , approximated by subgraphs of . For simplicity, we assume that both the metric and the induced Riemannian distance function are analytic. In addition, denote by for be the four straight segments of , and let boundary conditions be specified by , where is the union of a non-empty subset of .

Let be the cell decomposition of , consisting of four -cells placed at the corners of , four -cells, and one -cell. Let be a sequence of subdivisions by iteratively taking doubles, endowed with a local, consistent Riemannian structure and consistent combinatorial laplacians .

###### Theorem 3.1.

In the above setting, the determinant of the combinatorial laplacian has the following expansion:

 logdetΔi=1ϵ2i∫MF(g)dxdy+1ϵi∫∂MB(g)⋅ds+12log(ϵi)+O(1) (3.1)

where:

 F(gxx,gyy) =2πI⎛⎝\emph{Li}2(i√gyygxx)+\emph{Li}2⎛⎝i√gxxgyy⎞⎠⎞⎠−log(√gxxgyy) Bx(gxx,gyy) =−12log(√gxxgyy+√gxxgyy+gyy) (3.2) By(gxx,gyy) =−12log(√gxxgyy+√gxxgyy+gyy)

where is the dilogarithm function.

The approach to calculate the determinant is to study variations of the metric . Under a variation we have:

 δδglogdetΔiL=Tr((ΔiL)−1δΔiLδg) (3.3)

In the next section §4, we compute an estimate of the combinatorial Green’s function , which we use in section §5 to compute an estimate of (3.3). The asymptotic expansion (3.1) follows by integrating (3.3), with the constant of integration fixed by the expansion (1.2), corresponding to the case that .

We conclude this section by estimating the matrix elements of the combinatorial laplacians as .

#### Asymptotics of the Combinatorial Laplacian

First, we fix some useful notation borrowed from the finite difference calculus. For clarity, we leave implicit superscripts indicating the level of subdivision when there is no danger of confusion, for example for , for , etc. For with coordinates , define the finite difference operators:

 Dxf(x,y) =12(f(x+ϵ,y)−f(x−ϵ,y)) D2xf(x,y) =f(x+ϵ,y)−2f(x,y)+f(x−ϵ,y)

and similar operators in the -directions. These operators are consistent: for every , there are constants such that

 ∣∣∣1ϵDxIXf(x,y)−IX∂xf(x,y)∣∣∣≤ϵ2ηf∣∣∣1ϵ2D2xIXf(x,y)−IX∂2xf(x,y)∣∣∣≤ϵ2ηf (3.4)
###### Proposition 3.2.

Then the laplacian has the following asymptotic asymptotic expansion:

 ϵ2Δi=1√gyygxx⎛⎝√gyygxxD2x+√gxxgyyD2y+ϵ∂x√gyygxxDx+ϵ∂y√gxxgyyDy⎞⎠ (3.5) +O(ϵ2)D2x+O(ϵ2)D2y+O(ϵ3)Dx+O(ϵ3)Dy
###### Proof.

It is straightforward to check that on the square lattice, the combinatorial laplacian (2.1) can be written as a sum of finite difference operators and . (In fact, any operator that couples only nearest neighbor vertices and and annihilates constant functions can be written in terms of these finite difference operators). The proposition then follows from comparing the consistency of the finite difference operators (3.4), and consistency of the laplacian (2.2). ∎

## 4. Inverse of Combinatorial Laplacian

In this section, we derive an estimate for the combinatorial Green’s function . The approach can be summarized as follows. Since the combinatorial Laplacian is a consistent approximation of , the smooth Green’s function is an approximation of the combinatorial Green’s function where it is smooth. However, the Green’s function is logarithmically divergent where . We construct an approximate inverse matrix by replacing these logarithmic divergences with a lattice regularized logarithm function. We show that is a ”parametrix,” in the sense that for sufficiently large , the error

 R=I−ΔiGi

tends to zero, and for sufficiently large , the inverse can be written as

 (ΔiL)−1=Gi(I+R+R2+⋯) (4.1)

We begin first by recalling some facts about the Greens function of the smooth setting.

### 4.1. Smooth Green’s Function

Recall that when the Riemmanian distance function is smooth away from the diagonal, can be written neatly in terms of . Define as satisfying

1. for or

Then the Green’s function can be written:

 G(p1,p2)=−√g(p1)4πlog(d2(p1,p2))+ω(p1,p2) (4.2)

In this decomposition, both terms are divergent: the logarithmic term contains divergences as while contains divergences as .

To understand the divergent behaviour as , one can expand the distance function using the Hamilton-Jacobi equation. Letting , we calculate that as :

 d2(p1,p2)= gxxx2+gyyy2 (4.3) +12∂xgxxx3+12∂ygxxx2y+12∂xgyyxy2+12∂ygyyy3⋯

Then for the logarithm of the distance function:

 log(d2(p1,p2))=log(gxxx2+gyyy2)+log(1+O(|p1−p2|3)gxxx2+gyyy2) (4.4)

as . The term on the right is bounded, so we have isolated the divergence into the first term.

Turning to , recall that although for fixed , is smooth, as the function and its partial derivatives diverge. One can understand the behaviour of as using the method of images.(citation?). Define the reflected and the corner distance functions as:

 dRi(p1,p2)= inf{d(p1,b)+d(b,p2)|b∈∂Mi} (4.5) dRij(p1,p2)= inf{d(p1,b1)+d(b1,b2)+d(b2,p2) (4.6) |b1∈∂Mi,b2∈∂Mj%orb1∈∂Mj,b2∈∂Mi}

The reflected distance function is well defined for all when is sufficiently small, and similarly for the corner distance function. By definition, they satisfy when , and when . In addition, they satisfy the Hamilton-Jacobi equation, so that and .

Fix such that reflected distance function is well defined when is well defined for all and with . Let for all i, . For let . Redefine by the decomposition:

1. For :

 G(p1,p2)= −14πlog(d2(p1,p2))+14πlog(d2Ri(p1,p2)) +14πlog(d2Rj(p1,p2))−14πlog(d2Rij(p1,p2))+ω(p1,p2)
2. For :

 G(p1,p2)= −14πlog(d2(p1,p2))+14πlog(d2Ri(p1,p2))+ω(p1,p2)
3. For :

 Missing or unrecognized delimiter for \right

With this redefinition, it can be shown from a Schauder estimate that is smooth and bounded for all , so that the divergences have been pushed into the reflected distance functions.

We can compute the expansion of the reflected distance functions as before, using the Hamilton-Jacobi equations. Write the coordinates for , and for simplicity assume that is aligned with the y-axis, and is aligned with the x-axis. Then:

 d2Ri(p1,p2) =gxx(p1)(x1+x2)2+gyy(p1)(y2−y1)2+O(x21)+O(|p1−p2|2) d2Rij(p1,p2)