For $\Pi \u2282R2$, a connected, open, bounded set whose boundary is a finite union of disjoint polygons whose vertices have integer coordinates, the logarithm of the discrete Laplacian on $L\Pi \u2229Z2$ with Dirichlet boundary conditions has an asymptotic expression for large *L* involving the zeta-regularized determinant of the associated continuum Laplacian. When Π is not simply connected, this result extends to Laplacians acting on two-valued functions with a specified monodromy class.

## I. INTRODUCTION

*x*to

*y*. Note that for bounded Ω, this is a symmetric matrix with eigenvalues in (0, 4). The main focus of this paper is studying $det\Delta \u0303\Omega $ of the following form: we will fix a bounded open set $\Pi \u2282R2$ whose boundary is a disjoint union of polygons whose vertices are all in $Z2$, and consider Ω =

*L*Π for integer

*L*. Throughout this work, I will use Ω for a general domain and Π for a bounded domain of this polygonal form.

*x*−

*y*|

_{1}= 1 such that

*ρ*

_{xy}= 1/

*ρ*

_{yx}, we can define the discrete scalar Laplacian on Ω with Dirichlet boundary conditions and connection

*ρ*as

*ρ*appears in the form of products,

*y*

_{1}, …,

*y*

_{n+1}is a sequence that has no repeated elements except

*y*

_{n+1}≡

*y*

_{1}. These sequences can be naturally identified with simple closed curves; I will call the associated products monodromy factors. I will consider connections with the following property: for some finite $\Sigma \u2282\Omega c$, the monodromy factor is −1 if the associated closed curve winds around an odd number of elements of Σ and 1 otherwise (Fig. 2). From the above considerations, $det\Delta \u0303\Omega ,\rho $ is the same for all

*ρ*with this property for the same Σ, so we can let

*ρ*for any Σ. In fact, in this way, we obtain all of the connections that are “flat” (that is, for which the monodromy factor is 1 for all contractible curves) and where the monodromy factors take values only in {±1}.

These determinants are the partition functions of certain sets of essential cycle-rooted spanning forests.^{1} In the simply connected case (where Σ plays no role), the set of forests in question is simply the set of spanning trees on the graph formed by adding a “giant” vertex to $G(\Omega )$, which is connected to all the vertices on the boundary, and this identity is a restatement of the Kirchoff matrix-tree theorem. In this case and also for the case when Ω is doubly connected and Σ consists of a single point in the finite component of $\Omega c$, this partition function is, in turn, equal to the number of perfect matchings (configurations of the dimer model) on a “Temperleyan” graph constructed from $G(\Omega )$; however, for higher-genus planar domains, this correspondence involves a set of forests, which has a different characterization.^{2}

The main result of the present paper is the following theorem:

*as*

*L*→ ∞

*, where*

*#**denotes the**cardinality;***E**(*L*Π)*is the set of edges of**L*Π*,*|*e*|*is the length of the edge, and**α*_{1}(*e*)*depends only in the slope of**e*;**C**(*L*Π)*is the set of vertices (corners) of**L*Π*and**α*_{2}(*c*)*depends only on the opening angle of the corner and its orientation relative to the coordinate axes, or in other words, the slopes of the edges incident to**c**and whether the associated interior angle is acute or**obtuse;**α*_{3}(Π)*is the sum over corners of*Π*of*(*π*^{2}−*θ*^{2})/12*πθ*,*where**θ**is the angle measured in the interior of the corner;**and**α*_{4}(Π, Σ)*is the logarithm of the**ζ**-regularized determinant of the continuum Laplacian on a double cover of*Π*branched around*Σ*corresponding to the continuum limit of*$\Delta \u0303L\Pi $*, defined in Sec.*II*.*

It is known from calculations using other techniques that *α*_{0} = 4*G*/*π*, where *G* is Catalan’s constant; this can be verified from the expression in Eq. (102), but the details are unedifying, so I will not include them here. As far as I know, *α*_{1}, which is defined in Eq. (103), admits such an explicit expression only in special cases [for example, it follows from Eq. (4.23) of Ref. 3 that $\alpha 1(e)=\u221212log(2+1)$ when *e* is parallel to one of the coordinate axes].

*L*. In fact, the first term on the right-hand side of Eq. (5) is quadratic (by Pick’s theorem), but with linear and constant terms, which are generically nonzero and have no particular relationship to the other terms in the series, and

*∑*

_{C}

*α*

_{2}(

*C*) is independent of

*L*. As a result, while Theorem 1 implies that

*L*is a result of combining a number of bounds, which are helpful in different regimes and are clearly not themselves optimal. More concretely, the dominant contribution to the error term is expressed in terms of the probability of a rescaled Brownian bridge started at a point

*x*∈ Π coming within a small distance

*δ*of the boundary of Π without going further than

*δ*outside of Π; a more careful analysis of this problem would presumably lead to an improvement of the estimate in Eq. (5) and perhaps a modest simplification of the proof. If the convergence of the heat kernel were as fast as in the full-plane case ($O(1/n2)$, the error term would be improved to

*ζ*-regularized determinant of the Dirichlet Laplacian is a very well-studied object, not least because it gives a way of defining a (finite) partition function for free quantum field theories and has a very accessible relationship to the geometry of the domain on which it is defined; it has consequently been used to study the Casimir effect,

^{4}the effects of space–time curvature,

^{5}and the definition of conformal field theories.

^{6}In the case where Π is simply connected (i.e., a polygon), it has been described in great detail in Ref. 7. Among other things, the fact that

*α*

_{3}is the logarithm of such a determinant implies that

*α*

_{3}(

*L*Π,

*L*Σ) =

*α*

_{2}(Π) log

*L*+

*α*

_{3}(Π, Σ) [see Eq. (28)], so Eq. (5) is equivalent to

The form of this expression is related to the fact that *α*_{3} and *α*_{4} can be defined in terms of the continuum Laplacian; on the other hand, *α*_{0} and *α*_{1} are defined in terms of random walks on $Z2$ and so should depend on the details of the graph and (for *α*_{1}) the details of how the discrete boundary is chosen; *α*_{2} involves both discrete and continuous quantities. The leading term *α*_{0}, which is unambiguously defined and explicit, is known to correspond to the choice of $Z2$ as the underlying infinite graph. The boundary term is generically less explicit, but it is a sum of local terms, which depend only on the local shape of the boundary.

*α*

_{4}is the first term, which depends on Σ, for two different Σ on the same Π,

Expansions to this order for determinants of Laplacians (or dimer partition functions, or Ising model partition functions, which are closely related under suitable conditions) have now been studied for several decades. The presence of a term of constant order is interesting for applications to statistical physics, for example, since it is related to the notion of central charge in conformal field theory.^{9,10}

When comparing these results, it is important to note that boundary conditions (which, in the dimer model, are expressed in details of the geometry chosen) can have dramatic effects. For example, the first publication containing asymptotic formulas of this type^{11} considered the partition function of the dimer model on a discrete rectangle, which is not a Temperleyan graph, and so does not correspond to Dirichlet boundary conditions for the Laplacian; consequently, the expansion obtained there (which, in particular, has no logarithmic term) is not a special case of Theorem 1, although it can be placed in a common framework.

The aforementioned publication by Ferdinand in 1967 studied the discrete torus as well as a rectangle and noted the presence of a term of constant order expressed in terms of special functions. This expansion started from the expression of the partition function as a Pfaffian (square root of the determinant) or linear combination of Pfaffians of matrices, which had been explicitly diagonalized, and then carried out an asymptotic expansion using the Euler–Maclaurin formula. Subsequently, this approach was extended to other geometries, such as the cylinder and Klein bottle, and to obtain the asymptotic series to all orders in the domain size.^{12,13}

The first work to consider a case included in Theorem 1 was [see Ref. 3, in particular, Eq. (4.23)], who examined the determinant of the Laplacian on a rectangular lattice with Dirichlet boundary conditions (via the product of the nonzero eigenvalues of the Laplacian with Neumann boundary conditions on the dual lattice, adapting an earlier calculation,^{14} which was the first to include a logarithmic term); this was the first work to explicitly identify part of the expansion of the discrete Laplacian determinant with a zeta-regularized determinant. Subsequently, Kenyon^{15} used the result of this calculation (which was based on an exact diagonalization of the Laplacian), combined with a formula using dimer correlation functions to relate partition functions on different domains, to obtain a result for rectilinear polygons, that is, simply connected domains with sides parallel to the axes of $Z2$. Kenyon gives an alternative characterization of the term *α*_{3}, related to the limiting average height profile of the associated dimer model; recently, Finski^{16} showed that this is, in fact, the logarithm of the zeta-regularized determinant in a work, which also generalized the result to, among other things, nonsimply connected domains (including nontrivial connection), albeit still with the restriction that the boundary of the domain should always be parallel to one of the coordinate axes.

While revising this paper, I became aware of a recent study by Izyurov and Khristoforov,^{17} who, using an approach which overlaps to a large extent with my own, obtained a comparable result including some cases that are excluded by Theorem 1 and excluding some of the cases I consider. They consider a subset of an infinite periodic locally planar graph that is invariant under reflections (not necessarily $Z2$) and allow for Neumann or mixed Dirichlet–Neumann boundary conditions, as well as allowing the boundary to include punctures or slits. However, they consider only boundary components lying on lines of symmetry of the underlying graph, which (since the graph is periodic) means that the corner angles are restricted to be multiples of *π*/2 or *π*/3.

In the last two decades, there have also been a number of works relating the determinant of the Laplacian on higher-dimensional analogs of the discrete torus or rectangle^{18–20} to the corresponding zeta-regularized determinants. Although these authors introduced ideas that illuminate several aspects of the problem and make it possible to treat arbitrary dimension, they only treated graphs for which the spectrum of the Laplacian is given quite explicitly (although the authors of Ref. 18 do not use this directly, their treatment is based on the fact that the discrete torus is a Cayley graph, which is essentially what allows the spectrum to be calculated so explicitly). For some self-similar fractals, the problem has been approached using recursive relationships, which characterize the spectrum less explicitly,^{21} but this leads us into a rather different context.

As in Ref. 18, the proof of Theorem 1 is based on the relationship between the (matrix or zeta-regularized) determinant and the trace of the heat kernel, reviewed in Sec. II; however, once this relationship is introduced, I (like the authors of Ref. 17) treat the heat kernel using only a probabilistic representation, as the transition probability of a random walk or Brownian motion. In Sec. III, I introduce this representation (which is nonstandard only because of the presence of nontrivial monodromy factors) and present some straightforward bounds on the effect of changes in the domain Ω and the set Σ, which encodes the monodromy. Such bounds can be used to obtain an expansion for the behavior of the trace of the continuum heat kernel at small time (small, that is, on the scale associated with the domain); in Sec. IV, I review the proof of this expansion given in Ref. 22, which also provides the occasion to introduce certain definitions that will be used again later on to identify corresponding contributions in the discrete heat kernel. For times that are large on the lattice scale, the trace of the discrete heat kernel converges to that of the corresponding continuum heat kernel; in Sec. V, I give quantitative estimates on this convergence, based on the dyadic approximation. Then, in Sec. VI, I use a combination of the methods of the previous sections to control the behavior of the lattice heat kernel in a regime corresponding to that of the expansion for its continuum counterpart considered in Sec. IV. Finally, in Sec. VII, I combine all of these estimates to conclude the proof of Theorem 1.

Theorem 1 is limited to (1) planar regions, (2) polygons whose corners have integer coordinates, (3) Dirichlet boundary conditions, and (4) subsets of the square lattice, rather than more general planar graphs; let me comment on these limitations.

There does not seem to be any particular difficulty in extending the result to higher-dimensional polytopes, should such a result be of interest; however, several of the motivations for studying the problem are particular to the two-dimensional case. Similarly, it is quite straightforward to generalize this treatment to a torus, cylinder, Klein bottle, or Möbius strip, but these cases are all already well understood by other means.

^{12,13,23,24}Domains with punctures or slits^{25}are presumably accessible with some technical improvements; in fact, these have already been treated by a slightly different approach.^{17}The restriction to this class of polygonal domains is related to the construction of the term involving the “surface tension”

*α*_{1}in Eq. (5). Near the edges of the domain and away from the corners (the situation at the corners is more complicated, but in the end, the same considerations come into play), the domain coincides with an infinite half-plane in regions of size proportional to*L*so that the heat kernel there is very well approximated by the heat kernel on the intersection of $Z2$ with that half plane. The construction of*α*_{1}, given in Sec. VI, takes advantage of this and, in particular, the fact that because the slope of the boundary of the half-plane is rational, the result is a (singly-)periodic graph.The discrete heat kernel in Dirichlet boundary conditions has a particularly convenient probabilistic representation (see Sec. III for details): for a trivial connection (Σ = ∅), the diagonal elements are simply the probabilities that a standard random walk on $Z2$ returns to its starting point after a specified number of steps without ever having left the domain. This characterization has many helpful properties (most immediately, it is strictly monotone in the choice of domain). For a general connection, it is the expectation value of a function that is the monodromy factor of the trajectory of the random walk if it returns to its starting point without leaving the domain and zero otherwise, which is obviously bounded in absolute value by the probability appearing in the case of a trivial connection. The continuum heat kernel has a similar representation in terms of standard two-dimensional Brownian motion.

This is the key difference between my approach and that of Ref. 17. They are able to treat Neumann and, more importantly (since there is a duality relating Neumann and Dirichlet boundary conditions),

*mixed*boundary conditions by coupling the random walk under consideration to one on an infinite graph by reflecting the walk when it encounters a boundary component with Neumann boundary conditions. This requires, however, that the infinite graph be invariant under these reflections, which further limits the shapes of the domain, which can be considered as mentioned above.The restriction to the square lattice is most relevant for the technique I use to obtain quantitative control on the approximation between the discrete and continuum heat kernels in Sec. V. I use an approach based on the dyadic coupling of Komlós

*et al.*(KMT),^{26}which is essentially a one-dimensional technique, but which can be applied to random walks on $Zd$ and a few other graphs where the coordinates of the walker evolve independently. In fact, the KMT coupling can be used to obtain a similar control on the convergence of sums of I.I.D. sequences of vectors^{27}and thus random walks on infinite periodic graphs, as in Ref. 17. This is consistent with the similar expansions obtained in the last few years for periodic graphs embedded in the torus^{23}or Klein bottle,^{24}which have a similar structure, including a constant term that is universal (i.e., independent of many of the details of exactly which graph is chosen).It remains unclear whether a similar approach can be carried through for non-periodic graphs, which are needed to approximate many non-planar geometries. For graphs embedded in the torus, the asymptotic ratio corresponding to Eq. (9) is known to be the same for all graphs on which the simple random walk converges weakly to Brownian motion (a weaker notion than the one used here),

^{8}raising the question of whether a similar result holds for other geometries.

## II. HEAT KERNELS, ZETA FUNCTIONS, AND DETERMINANTS

*M*a symmetric

*m*×

*m*matrix with positive eigenvalues 0 <

*μ*

_{1}≤

*μ*

_{2}≤ …,

*μ*

_{m}, we define an entire function by

*s*> 0,

*ψ*(

*s*) ≔ Γ′(

*s*)/Γ(

*s*). As

*s*→ 0

^{+}, the two integrals become divergent at

*t*= 0, where Tr

*e*

^{−tM}→ Tr

*I*; however, these divergences cancel: we have

*s*→ 0

^{+}, since $1/\Gamma (s)=O(s)$ and $s\psi (s)=\u22121\u2212\gamma (s)+O(s2)$ [Ref. 28, Eq. (5.7.1, 5.7.6)], where

*γ*is the Euler–Mascheroni constant; note that the choice of

*e*

^{−γ}as the upper limit of integration gives the best possible cancellation, which is not strictly necessary but simplifies the resulting expressions. Since $Tre\u2212tM\u2212I=O(t)$ for

*t*→ 0, the remaining parts of the integrals on the right-hand side of Eq. (13) are convergent, leaving

*I*is the identity matrix on $\Omega \u2229Z2$, Eq. (15) becomes

To obtain comparable continuum entities, let Π_{Σ} denote a double cover of Π branched around each point in Σ and let Δ_{Π,Σ} be the Laplacian with Dirichlet boundary conditions acting on antisymmetric functions on Π_{Σ}, as detailed in Appendix A. For consistency with Eq. (1), we use the positive Laplacian $\Delta =\u2212\u22022\u2202x2\u2212\u22022\u2202y2$, giving a sign change and a factor of two with respect to many of the references cited.

_{Π,Σ}is discrete and bounded from below, and its density of states is asymptotically approximately linear- more precisely, in two dimensions, Weyl’s law states that the number

*N*(

*λ*) of eigenvalues of the Laplacian Δ

_{Ω}less than or equal to

*λ*(with multiplicity) satisfies

*s*> 1; as we shall shortly see, $\zeta \Delta \Pi ,\Sigma $ can be analytically continued to a neighborhood of 0, and—by analogy with Eq. (11)—$e\u2212\zeta \Delta \Pi ,\Sigma \u2032(0)$ is called the

*ζ*

*-regularized determinant*of the Laplacian.

*s*> 1 (as we shall see in Theorem 2, $e\u2212t\Delta \Pi ,\Sigma $ is trace class for

*t*> 0); the trace appearing in the integral has a noteworthy asymptotic expansion,

^{30}

^{,}

*a*

_{0}(Π) is 1/4

*π*times the area of Π,

*a*

_{1}(Π) is $\u22121/8\pi $ times the length of the boundary, and

*a*

_{2}(Π) is the sum over corners of $\pi 2\u2212\theta 2/24\pi \theta $, where

*θ*is the opening angle measured from the interior of Π (the first explicit formula for

*a*

_{2}appeared in Ref. 22; this simpler version, due to Ray, was first published in Ref. 31); an expansion for smooth manifolds with smooth boundaries (or no boundaries) is also known.

^{31,32}Although the equations in Refs. 22, 30, and 31 were formulated for the plane Laplacian or Laplace–Beltrami operator (equivalent to Σ = ∅), the proof of Eq. (19) in Ref. 22 extends straightforwardly to the dual cover. I will present a version of this proof in Sec. IV, both for completeness and because some of the entities introduced there are helpful for comparison with the discrete case.

*t*→ ∞, since the spectrum of Δ

_{Π,Σ}is bounded away from 0 [see Appendix A, especially Eq. (A1)], and so letting

*K*>

*e*

^{−γ}and Re

*s*> 1,

*K*→ ∞ to obtain

*F*and the various

*a*

_{n}(Π), we have

## III. HEAT KERNELS, BROWNIAN MOTION, AND RANDOM WALKS

*ρ*≡ 1, the elements of this matrix are simply the transition probabilities for

*n*steps of a random walk, which is killed on leaving Ω; to be precise,

*x*and $T\u0303X$, $X\u2282R2$, denotes the first time at which $W\u0303$ leaves

*X*, which in light of the definition Eq. (1), we understand to be the first time it jumps along a bond associated with a line segment not entirely contained in

*X*. For general

*ρ*,

*z*up to time

*n*, when

*ρ*is associated with some Σ, for the diagonal elements (which, ultimately, are the ones we are interested in) are given by

_{Ω,Σ}is just the usual Dirichlet Laplacian Δ

_{Ω}on Ω, and so for $t>0e\u2212t\Delta \Omega $ is the semigroup giving the solutions of the heat equation with Dirichlet boundary conditions. The Feynman–Kac formula then gives, for any suitable

*f*,

*x*,

*T*

_{Ω}is its first exit time from Ω, and

*y*∈ Ω), it is evident from this expression that

*P*

_{Ω}is a smooth function of

*y*for

*t*> 0, and so

*P*

_{Ω}.

The Laplacian acting on functions on a double cover with prescribed monodromy gives a counterpart to Eq. (29). Although the following statement is easily obtained by standard methods, it does not appear to be a readymade corollary of any results I was able to find, so I provide a proof in Appendix A:

*(note that the integral is over*Π

*, not the double cover*Π

_{Σ}

*), where*

*and*$W\Sigma (W;t)$

*denotes the sum over*

*y*∈ Σ

*of the winding number of*$W$

*around*

*y*

*in the time interval*[0,

*t*]

*, rounded to the nearest*

*integer.*

Note that the same limit is obtained without rounding off the winding number (something of the sort is needed to sensibly define values of *P*_{Π,Σ}(*x*, *y*; *t*) with *x* ≠ *y*, but these values are not important in the context of this article), but the rounded version is more convenient for the calculations I will present below.

This probabilistic representation can be used to prove a number of bounds. Let us begin with the following estimate on the effect of changes in Ω and/or Σ, which will be one of the basic ingredients for the rest of the present paper.

*For all*$\Omega ,\Theta \u2282R2$

*,*$\Sigma 1\u2282\Omega c$

*,*$\Sigma 2\u2282\Theta c$

*,*

*x*∈ Ω ∩ Θ

*,*

*R*≔ dist(

*x*, (Θ△Ω) ∪ (Σ

_{1}△Σ

_{2})). Using Eq. (39),

_{1}△Σ

_{2}before time

*t*, either of which requires it to travel a distance at least

*R*from its starting point; thus,

*B*

_{R}(

*x*) is the circle of radius

*R*centered at

*x*. The event of $W$ leaving

*B*

_{R}(

*x*) is contained in union of the events of moving by more than $22R$ in any one of the four cardinal directions, and the contribution of each of these events can be calculated using the reflection principle,

*P*, gives Eq. (38).□

For the lattice case, it is also possible to bound the effect of changes in the geometry in a similar fashion.

*There is a constant*

*C*

_{2}> 0

*such that for all*$\Omega ,\Theta \u2282R2$

*,*$\Sigma 1\u2282\Omega c$

*,*$\Sigma 2\u2282\Theta c$

*,*$x\u2208\Omega \u2229\Theta \u2229Z2$

*,*

*R*′ being the smallest integer such that $R\u2032\u226522R$. Noting that $W\u0303n\u2212W\u03030$ has the same distribution as $e\u03021+e\u030222Wn++e\u03021\u2212e\u030222Wn\u2212$ with $Wn\xb1$ two independent one-dimensional random walks started at 0,

## IV. REFERENCE GEOMETRIES FOR THE CONTINUUM HEAT KERNEL

In this section, we review the proof of Eq. (19). This is substantially the same proof as in Ref. 22, who considered the case of a planar region (i.e., without branching). We present it here in order to make it clear that it works unchanged for the more general case and to bring out a number of details in a way, which will make it easier to compare with the discrete version.

For any *r* > 0 and $x\u2208R2$, let *B*_{r}(*x*) denote the open disk of radius *r* centered at *x*. Let *κ* = *κ*(Π) > 0 be the such that for all *x* ∈ Π, *B*_{κ}(*x*) ∩ *∂*Π is contained in the union of two adjacent line segments of *∂*Π. We will approximate *P*_{Π}(*x*, *x*; *t*) (and, in due time, $P\u0303L\Pi (x,x;t)$) by the heat kernel in a reference geometry, a set *R*_{Π}(*x*) obtained by taking *B*_{κ}(*x*) ∩ Π and extending infinitely any components of *∂*Π present (see Fig. 3). Note that this gives either the whole plane, a half-plane, or an infinite wedge and that (up to translations) the same finite collection of wedges and half-planes appear for all $L\u2208N$.

*P*

_{R(x)}(

*x*,

*x*;

*t*) =

*P*

_{0}(

*t*) = 1/2

*πt*; note that

The boundary of Π is a disjoint union of semi-open line segments. Let $E$ be set whose elements are the semi-infinite rectangles swept out by taking one such line segment and displacing it perpendicularly in the direction of the interior of Π and removing the original line segment so that each $E\u2208E$ is closed on exactly one of its three sides (this is not immediately important, but when considering the discrete version, it will be helpful that this has the consequence that *LE*, for $L\u2208N$, is a disjoint union of *L* translates of *E*). For $E\u2208E$, let *ℓ*(*E*) denote the line segment used to construct *E* and *w*(*E*) denote the width of *E* (that is, the length of *ℓ*(*E*)) so that $\u2211E\u2208Ew(E)$ is the perimeter of Π; let *E*_{κ} ⊂ *E* be the *w*(*E*) × *κ* rectangle with side *ℓ*(*E*). Finally, let *H*(*E*) be the open half plane that contains *E* and whose boundary contains *ℓ*(*E*); then, for a given *E*, the set of *x* ∈ Π with *R*(*x*) = *H*(*E*) is a proper subset of *E*_{κ}. Some aspects of these definitions are illustrated in Fig. 4.

*x*

_{⊥}denote the component of

*x*perpendicular to

*ℓ*(

*E*), we have

*a*

_{1}as defined after Eq. (19). In addition,

Finally, let $C$ be the set of $C\u2282R2$, which are open infinite wedges matching the corners of Π. For such a *C*, let *φ*(*C*) be its opening angle, and let *E*^{1}(*C*), *E*^{2}(*C*) be the elements of $E$ associated with the line segments incident on the corner; thus, *C* = *H*(*E*^{1}(*C*)) ∩ *H*(*E*^{2}(*C*)) if *φ*(*C*) ≤ *π* and *C* = *H*(*E*^{1}(*C*)) ∪ *H*(*E*^{2}(*C*)) if *φ*(*C*) > *π* (Fig. 5). In addition, let *v*(*C*) be the vertex of *C*, and let *C*_{κ} be the set of *x* ∈ *C*, which are within a distance *κ* of both edges of *C* or, equivalently, the set of *x* ∈ Π for which *R*_{Π}(*x*) = *C*. For *j* = 1, 2, let $C\u0302\kappa j=(Ej)\kappa (C)\C$, and let $C\u0302j$ be the corresponding infinite wedge; both of these are ∅ if *φ*(*C*) ≥ *π*/2.

*E*

_{κ}integrals where

*x*∉ Π), where

*v*(

*C*) obtained by extending $E\kappa j$. $PC\u2220$ can be bounded using Theorem 3, and it satisfies $|PC\u2220(x;t)|\u22642C3t\u22121e\u2212k|x\u2212v(C)|2/t$ for $k=k(\Pi )=18minC\u2208C\u2061sin2\u2061\phi (C)/2$: for the first two cases in Eq. (52), this is straightforward; for the last case (without loss of generality assuming

*x*is closer to

*E*

_{2}than

*E*

_{1}, see Fig. 6), I write

*∂P*(

*x*;

*t*) for $x\u2208C\u03021\u222aC\u03022$. Consequently, we see that the integrals in

*x*, we see that this is, in fact, independent of

*t*, and by rotating and translating appropriately, we see that, in fact, it depends only on

*φ*(

*C*); an exact expression is given in Ref. 31. We also have

## V. LONG TIME: CONVERGENCE OF THE DISCRETE AND CONTINUOUS HEAT TRACES

*s*∈ [0,

*t*]; recall $P0(t)=PR2(x,x;t)=1/2\pi t$. Similarly, letting $B\u0303m$ be the process obtained by conditioning the random walk $W\u0303m$ to return to its starting point after 2

*n*steps,

*n*steps; representing this in terms of two independent 1

*D*walks as in Eq. (45),

*n*using Stirling’s formula.

*x*and

*y*may be different and the different time scales for the two processes, which are consistent with the asymptotic behavior of their variances for long times, and the factor of 1/2 in front of the discrete probabilities, which compensates for the reducibility of the discrete random walk.

This is helpful because the two bridge processes can be coupled using a variant of the dyadic coupling using a result of Ref. 33.

*There is a constant*

*C*

_{4}> 0

*such that for any*$y\u2208Z2$

*,*$x\u2208R2$,

*with*|

*x*−

*y*|

_{∞}≤ 1/2

*and any integer*

*n*> 1

*, there exists a coupling of*$Bs$

*and*$B\u0303s$

*such that*

The corresponding statement with *x* = *y* = 0 is obtained by rescaling (see Ref. 33, Corollary 3.2), which is, in turn, based on decomposing $B\u0303$ into two independent diagonal random walks as in the proof of Theorem 4 to use the one dimensional construction of Ref. 26; translating these processes to the desired starting points then gives the desired result up to a change in the constant, since |*x* − *y*| ≤ 1/2 ≤ log *n*.

□

*δ*> 0, restricting to the situation where the two processes remain within a distance

*δ*of each other for the whole time interval [0,

*n*], the difference on the right-hand side can be nonzero only if both $B$ and $B\u0303$ come within a distance

*δ*of $\Omega c$ but at least one of them remains entirely in Ω. This, in turn, requires that in the same time interval, $Bs$ leaves

*δ*=

*C*

_{4}log

*n*and noting that $|1\u22122P0(n)/P\u03030(2n)|\u2264C5/n$ [recall Eq. (60)], Eq. (61) gives

*x*from the boundary is small compared to $t$; fortunately, some distinctly suboptimal bounds (Lemmas 9 and 10 below) will suffice to give a serviceable estimate. The starting point is some estimates on hitting times, some of which are admittedly crude versions of well-known results, of which I provide basic proofs for completeness.

*For all*

*τ*> 0

*,*$\Omega \u2282R2$

*, and all*$x\u2208R2$

*,*

*with*

*C*

_{7}= 2

*π*/

*e*

*.*

*n*; as long as

*τ*≥

*e*|Ω|/2

*π*, we can always choose

*τ*, the bound is trivially correct.□

This can be used to obtain another estimate, which is more useful close to the boundary. I will state these estimates for domains of the following type:

*An* *S*(*ρ*) *set is an open set* $\Omega \u2282R2$ *such that any point* $y\u2208R2\\Omega $ *is an endpoint of a line segment* *ℓ*_{y} *of length* *ρ*, *which does not intersect* Ω*.*

Note that the polygon Π we consider is an *S*(1) set and *L*Π is an *S*(*L*) set; Π^{+}(*δ*) and Π^{−}(*δ*) [respectively, (*L*Π)^{+}(*δ*) and (*L*Π)^{−}(*δ*) are *S*(1/2) (respectively, *S*(*L*/2)] for *δ* < *C*_{8} (respectively, *δ*/*L* < *C*_{8}) for some Π-dependent *C*_{8}.

*There exists*

*C*

_{9}> 0

*such that if*

*τ*> 0

*,*Ω

*is an*$S(\tau )$

*set, and*

*x*∈ Ω

*, then*

*with*

*R*

*being the distance between*

*x*

*and*

*∂*Ω

*, as long as*$R\u2264e\u22122/e\tau $

*.*

*y*be a point in

*∂*Ω with |

*x*−

*y*| =

*R*, and let

*ℓ*

_{y}be the line segment of length $\tau $ associated with

*y*by Definition 7. Since $\u2113\u2282\Omega c$, trivially

*ρ*> 0. For $\rho \u2264\tau $, the second term on the right-hand side is the solution of a harmonic problem, which can be solved explicitly: for

*z*a unit complex number depending on

*ℓ*and

*y*,

This can then be use to obtain an estimate on the heat kernel, which will be useful for estimating $P(L\Pi )+(\delta )$, and hence the difference, near the boundary where any Brownian motion is very likely to exit (*L*Π)^{+}(*δ*):

*There exists*

*C*

_{10}> 0

*such that if*

*t*> 0

*,*Ω

*is an*$S(\tau )$

*set for some*

*τ*≤

*t*/3

*and*

*x*∈ Ω

*, then*

*with*

*R*

*being the distance between*

*x*

*and*

*∂*Ω

*, as long as*$R\u2264e\u22122/e\tau $

*.*

*τ*≤

*t*/3 in the last inequality, the result follows from Lemma 8.□

When the starting point is far from the boundary, I can instead obtain a useful estimate via the probability of a Brownian motion, having left (*L*Π)^{−}(*δ*), manages to travel far enough to get back to its starting point without leaving (*L*Π)^{+}(*δ*), whose boundary is now at a distance of order *δ*.

*There exists*

*C*

_{11}

*for which the following holds. For any*

*S*(

*ρ*),

*set*Ξ

*, any open measurable*Ω ⊂ Ξ

*with*max

_{y∈∂Ω}min

_{z∈∂Ξ}|

*y*−

*z*| =

*δ*′

*, and any*

*x*∈ Ω

*with*dist(

*x*,

*∂*Ω) =

*R*

*,*

*T*

_{Ω}and $WT\Omega $,

*B*

_{q}(

*z*), which contains

*y*but not

*x*before exiting Ξ,

*z*be a point on the boundary of Ξ with |

*z*−

*y*| ≤

*δ*′, for all

*q*≤

*ρ*, we can use the assumption that Ξ is an

*S*(

*ρ*) set as in Eq. (72) to bound $PyTBq(z)\u2264T\Xi \u2264(4/\pi )q/\delta \u2032$, and noting

*P*(

*w*,

*x*;

*u*) ≤ 1/(

*eπ*|

*w*−

*x*|

^{2}) ≤ 1/(

*eπ*(

*R*−

*q*)

^{2}), Eq. (76) follows by choosing

*q*= min(

*ρ*,

*R*/2).□

*There exist*

*C*

_{12},

*C*

_{13},

*C*

_{14},

*C*

_{15}> 0

*, depending on*Π

*, such that for*

*δ*≤

*C*

_{12}

*L*

*and*dist(

*x*,

*∂*Π

^{+}(

*δ*)) =

*R*

*,*

*whenever*$C14\u2264t\u226434L2$

*, and*

*when*$34L2\u2264t\u2264C15L5/2$

*.*

First of all, there exist *C*_{12}, *C*_{16} < *∞* such that for all *δ* ≤ *C*_{12}, Π^{+}(*δ*) is an *S*(1/2) set and $maxy\u2208\u2202\Pi \u2212(\delta )minz\u2208\u2202\Pi +(\delta )|y\u2212z|\u2264C16\delta $; rescaling, this implies that (*L*Π)^{+}(*δ*) is *S*(*L*/2) and $maxy\u2208\u2202(L\Pi )\u2212(\delta )minz\u2208\u2202(L\Pi )+(\delta )|y\u2212z|\u2264C16\delta $ for *δ* ≤ *C*_{12}*L*.

Then, by choosing *C*_{14} large enough, Lemma 9 applies in the first case on the right-hand side of Eq. (79) with *τ* = *t*/3, giving the advertised bound, and for *C*_{15} small enough likewise for the first case in Eq. (80) with *τ* = *L*^{2}/4. The other cases follow from Lemma 10 with *ρ* = *L*/2.□

Integrating in *x*, we obtain the following corollary:

*There exists*

*C*

_{17}

*such that*

*whenever*

*δ*≤

*C*

_{12}

*L*

*and*

*C*

_{14}≤

*t*≤

*C*

_{15}

*L*

^{5/2}

*.*

We now arrive at the conclusion of this section:

*For each*Π

*, there exists*

*C*

_{18}< ∞

*such that whenever*$K\u2208N$

*and*

*e*

^{9/2}≤

*K*≤

*L*

*,*

*δ*=

*C*

_{4}log

*n*. For

*n*≤

*C*

_{15}

*L*

^{5/2}, we can bound the first integral using Corollary 12 and the second using $P(L\Pi )+(\delta )(x,x;n)\u22641/4\pi n$, giving

*n*>

*C*

_{15}

*L*

^{5/2}, a stronger bound follows easily from Lemma 8; the error made by replacing the

*t*integral in Eq. (82) with a sum can easily be seen to be $O(L2/K4)$ using the Euler–Maclaurin formula, and so Eq. (82) follows by summing the above estimates.□

## VI. THE REFERENCE GEOMETRY TERMS IN THE DISCRETE CASE

*α*

_{0},

*α*

_{1},

*α*

_{2}in Eq. (5). To take into account the presence of a lattice scale, we define

*a*

_{0}(Π),

*a*

_{1}(Π),

*a*

_{2}(Π); however, their time dependence is not so simple as in the continuum case. For

*H*a half-plane, we let

*A*

_{2}is actually independent of

*L*,

*A*

_{2}(

*L*Π,

*n*) =

*A*

_{2}(Π,

*n*), since rescaling each wedge $C\u2208C$ by an integer factor gives the same result; it is also invariant under the symmetries of $Z2$, and in fact, $A\u03022(C,n)$ is evidently determined by the opening angle of

*C*and its orientation relative to the lattice. Analogously,

*A*

_{1}(

*L*Π,

*n*) =

*LA*

_{1}(Π,

*n*), and $A\u03021(E,L,n)$ is fixed by the length and orientation of the edge of

*L*Π corresponding to

*E*.

*κL*from the points of interest, which can be bounded using Theorem 4, giving the counterpart of Eq. (56),

*n*≤

*L*

^{2}. Applying Theorem 4 again to bound the difference between $P\u0303L\Pi R$ and $P\u0303L\Pi $ gives an estimate of similar magnitude,

*K*≤

*L*.

*A*

_{0}, this is quite simple: recalling Eq. (46),

*A*

_{1}, recalling Eq. (49), we have

*H*(

*E*); as in Eq. (66), we can bound this difference using Theorem 5, giving

*n*> 1, |

*x*−

*y*|

_{∞}≤ 1/2, with

*δ*=

*C*

_{4}log

*n*. This difference can be bounded as in Lemma 11 using Lemmas 8 to 10, with the simplification that the complement of [

*H*(

*E*)]

^{+}(

*δ*) always contains an infinite half-line touching any point on its boundary (in terms of Definition 7, it is

*S*(

*ρ*) for any

*ρ*> 0), giving

*n*

^{2}, which does not depend on the distance from the boundary; however, it does give

*E*

_{κ}(cf. Fig. 4), with

*R*

_{1}being any number such that $|ER1|=#ER1\u2229Z2$. Further away from the edge, we can use Theorems 3 and 4 to bound

*∂P*

_{H(E)}and $\u2202P\u0303H(E)$ separately, and setting

*R*

_{1}as close as possible to

*n*

^{19/14}, the contribution from $E\ER1$ is superexponentially small for large

*n*. In conclusion, we have

*n*> 1; note, in particular, that this decays faster than $1/n$ for

*n*large.

*v*(

*C*), by the same reasoning as in the derivation of Eq. (55); so with

*R*

_{1}=

*n*

^{25/28}, we have

*n*large enough.

*e*an edge of

*L*Π,

*E*

_{e}the associated element of $E(L\Pi )$,

*c*a corner, and

*C*

_{c}the associated wedge in $C(L\Pi )$. Note that in light of the properties noted after Eq. (90), these definitions have the properties specified in points 2 and 3 of Theorem 1, and furthermore,

*A*

_{2}.

## VII. CONCLUDING THE PROOF OF THEOREM 1

*L*/

*K*→

*∞*, while in Theorem 13, we saw that

*K*,

*L*/

*K*,

*K*

^{9/7}/

*L*→

*∞*. All of the error terms will ultimately have similar forms, so I will set $K=ceil(C34L/logL)$ with

*C*

_{34}chosen large enough so that the exponential error terms are negligible; we then have

*L*→

*∞*.

*F*

_{Π}in Eq. (20) and the relationship between

*P*

_{Π,Σ}and $Tr\u2009e\u2212t\Delta \Pi ,\Sigma $ (Theorem 2),

*K*chosen as discussed above. Rescaling

*F*as in Eq. (27) and taking advantage of the small

*t*asymptotics in Eq. (19), we have

*α*

_{0},

*α*

_{1}, and

*α*

_{2}using Eq. (105) and $\alpha 4(L\Pi ,L\Sigma )=\zeta \Delta L\Pi ,L\Sigma \u2032(0)$ using Eq. (26), we obtain Eq. (8). As mentioned before, Theorem 1 then follows by using Eq. (28).

## ACKNOWLEDGMENTS

I thank Bernard Duplantier for introducing me to this problem and pointing out its relationship to the heat kernel. The outline of the proof occurred to me during the workshop “Dimers, Ising Model, and their Interactions” at the Banff International Research Station, and I thank the organizers for the invitation to attend the workshop as well as the many participants in the discussions which provoked it. I would also like to thank Alessandro Giuliani for discussing the work with me through all the stages of development and the anonymous referee whose comments led to a considerable number of corrections and improvements.

This work, a major part of which was conducted at the Department of Mathematics and Physics of the Università degli Studi Roma Tre (Rome, Italy), was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (ERC CoG UniCoSM, Grant Agreement No. 724939 and ERC StG MaMBoQ, Grant Agreement No. 802901). The final stages were also partially supported by the MIUR Excellence Department Project MatMod@TOV awarded to the Department of Mathematics, University of Rome Tor Vergata.

## AUTHOR DECLARATIONS

### Conflict of Interest

The author has no conflicts to disclose.

### Author Contributions

**Rafael L. Greenblatt**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

### APPENDIX A: THE LAPLACIAN WITH DIRICHLET BOUNDARY CONDITIONS ON A DOUBLE COVER

Let Π_{Σ} denote a double cover of Π branched around each point in Σ, and let *p*: Π_{Σ} → Π be the associated projection map. Let *L*^{2}(Π, Σ) be the set of equivalence classes (with respect to almost-everywhere equality) of square-integrable functions $f:\Pi \Sigma \u2192R$ with the property that *f*(*x*) = −*f*(*y*) if *p*(*x*) = *p*(*y*) but *x* ≠ *y*. *L*^{2}(Π, Σ) is evidently a closed linear subspace of *L*^{2}(Π_{Σ}) and so is itself a Hilbert space with the same inner product. Let $C0\u221e(\Pi ,\Sigma )$ be the set of smooth functions *f* ∈ *L*^{2}(Π, Σ) such that for all *y* ∈ *∂*Π_{Σ}, lim_{x→y} *f*(*x*) = 0 and the corresponding limits of all of the derivatives of *f* exist.

_{Π,Σ}is defined as the Friedrichs extension of the (positive) Laplacian −∇ · ∇ on $C0\u221e$, which is self-adjoint with respect to the inner product of

*L*

^{2}(Π, Σ). It is a strictly positive operator: letting Δ

_{Π}be the Dirichlet Laplacian on Π (not the double cover), for any $f\u2208C0\u221e(\Pi ,\Sigma )$, the function $f\u0303$ defined by $f\u0303(p(x))=|f(x)|$ is in the domain of Δ

_{Π}; we can extend it to a function in the domain of Δ

_{R}for

*R*a rectangle containing Π, and then,

*L*

_{2}inner product; of course, the spectrum of Δ

_{R}is explicitly known and strictly positive, so the last ratio is uniformly bounded from below by a positive number, a bound that thus extends to all

*f*in the domain of Δ

_{Π,Σ}. Among other things, this combines with the spectral theorem to imply that $e\u2212t\Delta \Omega ,\Sigma $ is defined for all

*t*≥ 0.

*x*∈ Π

_{Σ}. Consider the family of operators

*K*

_{t}on

*L*

^{2}(Π, Σ) defined by

_{Π,Σ}acts locally on smooth functions like the ordinary Laplacian, Itô’s formula implies that

*L*

^{2}(Π, Σ), and evidently,

*K*

_{0}is the identity; in other words,

*K*

_{t}is a one-parameter group generated by $\u221212\Delta \Pi ,\Sigma $; there is only one such group (see Ref. 34, Theorem 1.7), so $Kt=e\u2212(t/2)\Delta \Pi ,\Sigma $.

*For any closed, simply connected*

*B*⊂ Π

*, there is a function*

*Q*

_{B}(

*x*,

*y*;

*t*)

*from*Π

_{Σ}×

*p*

^{−1}(

*B*) × (0,

*∞*)

*to*$R$

*such that*

*for any*

*g*∈

*L*

^{2}(Π, Σ)

*with*

*supp g*⊂

*B*

*; furthermore,*

*Q*

_{B}

*is*

*C*

^{∞}

*as a function of*

*y*

*, and*

*where*

*P*

*is the full-plane heat kernel introduced in*

*Eq. (33)*

*.*

*B*

_{1},

*B*

_{2}⊂ Π be open and simply connected with $B\u03041\u2282B\u2282B2$ and

*δ*≔

*d*(

*B*,

*∂B*

_{1}) ∨

*d*(

*B*

_{1},

*∂B*

_{2}) > 0. Letting $W$ denote a planar Brownian motion started at

*x*, I introduce two related sequences of hitting times as follows: let

*τ*

_{0}=

*σ*

_{0}= 0, and for

*j*= 1, 2, …,

*B*

_{1}and the boundary of

*B*

_{2}),

*J*≔ max{

*j*= 0, 1, …:

*σ*

_{j}≤

*t*} is almost surely finite. Furthermore,

*B*

_{2}is simply connected, thanks to the indicator function,

*G*(

*y*;

*s*) is an integral over paths, which remain in the same connected component of

*p*

^{−1}(

*B*

_{2}) as $y=W\sigma J\Sigma $. To spell this out, I introduce the following notation: Let $B21$ and $B22$ be the two connected components of

*B*

_{2},

*P*are non-negative. Note also that the last expression in Eq. (A14) is an integral over values of

*P*(

*y*,

*p*(

*z*);

*s*) with either

*s*=

*t*(if

*σ*

_{J}= 0) or

*y*∈

*∂B*

_{1}∪

*∂B*

_{2}[and thus |

*y*−

*p*(

*z*)|≥

*δ*]. With the other arguments so restricted,

*P*is uniformly smooth as a function of

*z*, and so we also see that

*Q*

_{B}is smooth as claimed.□

Given that they are smooth in the second argument, any two kernels *Q*_{B}(*x*, ⋅; *y*) and *Q*_{B′}(*x*, ⋅; *t*) defined on different domains coincide on the intersection of their domains, and so we can extend by linearity to the rest of *L*^{2}(Π, Σ) and obtain the following corollary:

*There is a function*$Q\Pi ,\Sigma :\Pi \Sigma \xd7\Pi \Sigma \xd7(0,\u221e)\u2192R$

*such that*

*for any*

*f*∈

*L*

^{2}(Π, Σ);

*furthermore,*

*Q*

_{Π,Σ}

*is*

*C*

^{∞}

*as a function of*

*y*,

*and*

With this result in hand, I am ready to present the following proof:

*U*⊂ Π, let $fU:\Pi \Sigma \u2192R$ be either of the two functions, which is equal to 1 on one of the connected components of

*p*

^{−1}(

*U*), −1 on the other, and zero elsewhere. Evidently,

*f*

_{U}∈

*L*

^{2}(Π, Σ), (

*f*

_{U},

*f*

_{U}) = 2|

*U*|, and

*f*

_{U}was chosen. Note also that if diam

*U*is small enough, in the last integral, $\gamma (W0\Sigma ,Wt\Sigma )$ is equivalent to $ei\pi W\Sigma (W;t)$ (recall that

*W*

_{Σ}was defined in the statement of Theorem 2 to take integer values). Examining Eq. (37) in light of this, we see that for

*x*∈ Π

_{Σ}, the definition of

*P*

_{Π,Σ}there gives

*Q*

_{Π,Σ}shown in Corollary 15.

*N*a positive integer, denote the nonempty sets that coincide with a connected component of a set of the form $\Pi \u22292\u2212N[m,m+1)\xd7[n,n+1)$ for some $m,n\u2208Z$; then, $UN$ is a finite set of disjoint, simply connected Borel sets whose union is Π. Numbering the elements of each $UN$ arbitrarily so that

*L*

^{2}(Π, Σ), and by applying the Hilbert–Schmidt process to this sequence, we obtain an orthonormal basis

*ψ*

_{1},

*ψ*

_{2}, … such that

*N*. Since $e\u2212t\Delta \Pi ,\Sigma $ is non-negative, this suffices to show that

*U*

_{N}(

*x*) is the unique element of $UN$, which includes

*x*. As

*N*→

*∞*, the quantity in brackets converges pointwise to the same limit as in Eq. (A19) with

*t*replaced by 2

*t*, i.e., to

*P*

_{Π,Σ}(

*p*(

*x*),

*p*(

*x*); 2

*t*), and this together with the bound in Eq. (A16) provide the conditions to apply the dominated convergence theorem and obtain

*x*∈ Π, is equivalent to Eq. (36).□

### APPENDIX B: A SIMPLE BOUND ON THE DISTRIBUTION OF A SIMPLE RANDOM WALK

*For all integers*

*n*≥ 1

*and*0 ≤

*m*≤

*n*

*,*

*m*= 0 or

*m*=

*n*, the bound is obvious, so we consider 1 ≤

*n*≤

*n*− 1. In this case, using Stirling’s formula with explicit error estimates,

^{35}

*L*

_{n}(

*μ*) = (

*μ*+ 1/2) log

*μ*+ (

*n*−

*μ*+ 1/2) log(

*n*−

*μ*), we have $Ln\u2032(n/2)=0$,

*n*/2 is the unique local minimum of $Ln\u2032\u2032(\mu )$. It is easy to see that

*n*large enough (in fact, for

*n*≥ 4), and so we have

*n*, and the result can be trivially extended to smaller

*n*.□

## REFERENCES

*Quantum Fields and Strings: A Course for Mathematicians*

*Methods of Mathematical Physics*

*Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem*

*One-parameter Semigroups*