For ΠR2, 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 LΠZ2 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.

For a domain ΩR2, let G(Ω) be the graph whose vertex set is Ω̃ΩZ2 and whose edge set E(Ω) is the set of pairs x,yΩ̃ such that the line segment xȳ has length one and is entirely contained in Ω (see Fig. 1 for examples). The discrete Laplacian on Ω with Dirichlet boundary conditions is the operator Δ̃Ω on 2(ΩZ2) given by
where xȳ is the closed line segment from 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 detΔ̃Ω of the following form: we will fix a bounded open set ΠR2 whose boundary is a disjoint union of polygons whose vertices are all in Z2, and consider Ω = LΠ for integer L. Throughout this work, I will use Ω for a general domain and Π for a bounded domain of this polygonal form.
FIG. 1.

Example of a shape Π and the graphs G(LΠ) for different integer values of L (not to scale).

FIG. 1.

Example of a shape Π and the graphs G(LΠ) for different integer values of L (not to scale).

Close modal
FIG. 2.

Example of a region Π and set Σ=σ1,σ2 used to specify monodromy factors; in this example, the solid curves have monodromy 1 and the dashed curves have monodromy −1, independent of their orientation.

FIG. 2.

Example of a region Π and set Σ=σ1,σ2 used to specify monodromy factors; in this example, the solid curves have monodromy 1 and the dashed curves have monodromy −1, independent of their orientation.

Close modal
In fact, I will consider a somewhat more general Laplacian, sometimes called the twisted Laplacian. For an assignment of ρxyC to each x,yΩZ2 with |xy|1 = 1 such that ρxy = 1/ρyx, we can define the discrete scalar Laplacian on Ω with Dirichlet boundary conditions and connection ρ as
Writing out detΔ̃Ω,ρ by the Leibniz formula, ρ appears in the form of products,
where y1, …, yn+1 is a sequence that has no repeated elements except yn+1y1. 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 ΣΩc, 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, detΔ̃Ω,ρ is the same for all ρ with this property for the same Σ, so we can let
for an arbitrary chosen such connection; it is easy to construct such a 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(Ω), 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 Ωc, 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(Ω); 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:

Theorem 1.
as L → ∞, where
  1. # denotes the cardinality;

  2. E(LΠ) is the set of edges of LΠ, |e| is the length of the edge, and α1(e) depends only in the slope of e;

  3. C(LΠ) is the set of vertices (corners) of LΠ and α2(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;

  4. α3(Π) is the sum over corners of Π of (π2θ2)/12πθ, where θ is the angle measured in the interior of the corner; and

  5. α4(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 Δ̃LΠ, defined in Sec. II.

It is known from calculations using other techniques that α0 = 4G/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, α1, 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 α1(e)=12log(2+1) when e is parallel to one of the coordinate axes].

Note that Eq. (5) is written as a mixture of terms involving the geometry of LΠZ2 as a graph with a boundary and terms involving the limiting shape Π, rather than an expansion in orders of 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 Cα2(C) is independent of L. As a result, while Theorem 1 implies that
the constant term in this expression is not generically given by the logarithm of the zeta-regularized determinant without further corrections.
The order of the error term is mainly fixed by the estimates in Secs. V and VI on the speed of convergence of the discrete heat kernel with Dirichlet boundary conditions to its continuum counterpart; the peculiar dependence is on 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
which is probably the highest precision, which can be studied by the methods presented here.
The 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 α3 is the logarithm of such a determinant implies that α3(LΠ, LΣ) = α2(Π) log L + α3(Π, Σ) [see Eq. (28)], so Eq. (5) is equivalent to
which I, in fact, prove first.

The form of this expression is related to the fact that α3 and α4 can be defined in terms of the continuum Laplacian; on the other hand, α0 and α1 are defined in terms of random walks on Z2 and so should depend on the details of the graph and (for α1) the details of how the discrete boundary is chosen; α2 involves both discrete and continuous quantities. The leading term α0, 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.

One consequence of Eq. (5) is that since α4 is the first term, which depends on Σ, for two different Σ on the same Π,
a similar result was shown for a variety of toric graphs (not necessarily Z2, or even periodic) in Ref. 8.

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 type11 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, Kenyon15 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 α3, related to the limiting average height profile of the associated dimer model; recently, Finski16 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 rectangle18–20 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.

  1. 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 slits25 are presumably accessible with some technical improvements; in fact, these have already been treated by a slightly different approach.17 

  2. The restriction to this class of polygonal domains is related to the construction of the term involving the “surface tension” α1 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 α1, 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.

  3. 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.

  4. 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 vectors27 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 torus23 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.

For M a symmetric m × m matrix with positive eigenvalues 0 < μ1μ2 ≤ …, μm, we define an entire function by
this definition has the consequence that
or, equivalently, detM=expζM(0); to evaluate this, we note that for Re s > 0,
ζM(s)=j=1m1Γ(s)0ts1etμjdt=1Γ(s)0ts1Tr etMdt,
where Γ(s)0ts1etdt, and consequently,
where ψ(s) ≔ Γ′(s)/Γ(s). As s → 0+, the two integrals become divergent at t = 0, where Tr etM → Tr I; however, these divergences cancel: we have
for s → 0+, since 1/Γ(s)=O(s) and sψ(s)=1γ(s)+O(s2) [Ref. 28, Eq. (5.7.1, 5.7.6)], where γ is the Euler–Mascheroni constant; note that the choice of eγ as the upper limit of integration gives the best possible cancellation, which is not strictly necessary but simplifies the resulting expressions. Since TretMI=O(t) for t → 0, the remaining parts of the integrals on the right-hand side of Eq. (13) are convergent, leaving
In our case, we are therefore interested in TretΔ̃Ω,ρ, which is related to a continuous time random walk. It will mostly be easier to consider the corresponding discrete-time random walk, which can be obtained through the following manipulations. Noting
where I is the identity matrix on ΩZ2, Eq. (15) becomes
ζΔ̃Ω,ρ(0)=0e4t1[0,eγ](t)dttTr I+n=14n!0e4t(4t)n1dtTr (I14Δ̃Ω,ρ)n=(log4)#ΩZ2+n=112nTr (I14Δ̃Ω,ρ)2n
(note that in the last step I have omitted the traces of odd powers, which vanish).

To obtain comparable continuum entities, let ΠΣ denote a double cover of Π branched around each point in Σ and let ΔΠ,Σ be the Laplacian with Dirichlet boundary conditions acting on antisymmetric functions on ΠΣ, as detailed in  Appendix A. For consistency with Eq. (1), we use the positive Laplacian Δ=2x22y2, giving a sign change and a factor of two with respect to many of the references cited.

The spectrum of ΔΠ,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 ΔΩ less than or equal to λ (with multiplicity) satisfies
it is usually stated for the Laplacian on functions on planar domains, but the well-known variational proof (see Ref. 29, Sec. 6.4) also applies to functions on double covers—so the zeta function
ζΔΠ,Σ(s)λspec ΔΠ,Σλs
is well defined and analytic for Re s > 1; as we shall shortly see, ζΔΠ,Σ can be analytically continued to a neighborhood of 0, and—by analogy with Eq. (11)—eζΔΠ,Σ(0) is called the ζ-regularized determinant of the Laplacian.
The analogs of Eqs. (12) and (13) hold for Re s > 1 (as we shall see in Theorem 2, etΔΠ,Σ is trace class for t > 0); the trace appearing in the integral has a noteworthy asymptotic expansion,30,
where a0(Π) is 1/4π times the area of Π, a1(Π) is 1/8π times the length of the boundary, and a2(Π) is the sum over corners of π2θ2/24πθ, where θ is the opening angle measured from the interior of Π (the first explicit formula for a2 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.
TretΔΠ,Σ also decreases exponentially as t → ∞, since the spectrum of ΔΠ,Σ is bounded away from 0 [see  Appendix A, especially Eq. (A1)], and so letting
it is usually stated for the Laplacian on functions on planar domains, but the well-known variational proof (see Ref. 29, Sec. 6.4) also applies to functions on double covers}, so the zeta function
we obtain, for any K > eγ and Re s > 1,
we thus have
and we can analytically continue and then take the limit K → ∞ to obtain
for all |s|<12, and noting that
along with
lims01Γ(s)=0,lims0dds1Γ(s)=1,lims01sΓ(s)=1, and ,lims0dds1sΓ(s)=γ
Note, finally, that from the definitions of F and the various an(Π), we have
I now turn to the representation I will use to estimate the trace of (I14Δ̃Ω,ρ)n appearing in Eq. (17). When 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,
where W̃ is a simple random walk on Z2 starting at x and T̃X, XR2, denotes the first time at which W̃ 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,
Letting Wz(W̃;n) denote the winding number (index) of W̃ around 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
with WΣ(W̃;n)σΣWσ(W̃;n).
Let us now look at the continuum. With Σ = ∅, ΔΩ,Σ is just the usual Dirichlet Laplacian ΔΩ on Ω, and so for t>0etΔΩ is the semigroup giving the solutions of the heat equation with Dirichlet boundary conditions. The Feynman–Kac formula then gives, for any suitable f,
where W is a two-dimensional Brownian motion starting from x, TΩ is its first exit time from Ω, and
is the full-plane heat kernel; in other words, et2ΔΩ has the integral kernel
Apart from the boundary (i.e., for y ∈ Ω), it is evident from this expression that PΩ is a smooth function of y for t > 0, and so
which is helpful in deriving estimates on PΩ.

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:

Theorem 2.
(note that the integral is over Π, not the double cover ΠΣ), where
and WΣ(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 PΠ,Σ(x, yt) with xy, 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.

Theorem 3.
For all Ω,ΘR2, Σ1Ωc, Σ2Θc, x ∈ Ω ∩ Θ,

For brevity, let R ≔ dist(x, (Θ△Ω) ∪ (Σ1△Σ2)). Using Eq. (39),
For the difference in the expectation to be nonzero, W must either enter Θ△Ω or wind around a point in Σ1△Σ2 before time t, either of which requires it to travel a distance at least R from its starting point; thus,
where BR(x) is the circle of radius R centered at x. The event of W leaving BR(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,
and inserting this in Eq. (40), together with the exact expression for 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.

Theorem 4.
There is a constant C2 > 0 such that for all Ω,ΘR2, Σ1Ωc, Σ2Θc, xΩΘZ2,

Recalling Eq. (31),
and as in the proof of Theorem 3, this can be bounded as
with R′ being the smallest integer such that R22R. Noting that W̃nW̃0 has the same distribution as ê1+ê22Wn++ê1ê22Wn with Wn± two independent one-dimensional random walks started at 0,
which implies the announced result together with Lemma 16.□

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 xR2, let Br(x) denote the open disk of radius r centered at x. Let κ = κ(Π) > 0 be the such that for all x ∈ Π, Bκ(x) ∩ Π is contained in the union of two adjacent line segments of Π. We will approximate PΠ(x, xt) (and, in due time, P̃LΠ(x,x;t)) by the heat kernel in a reference geometry, a set RΠ(x) obtained by taking Bκ(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 LN.

FIG. 3.

Example of a polygonal domain Π showing the regions for which RΠ(x) is of the specified form.

FIG. 3.

Example of a polygonal domain Π showing the regions for which RΠ(x) is of the specified form.

Close modal
When R(x)=R2, we have PR(x)(x, xt) = P0(t) = 1/2πt; 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 EE 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 LN, is a disjoint union of L translates of E). For EE, let (E) denote the line segment used to construct E and w(E) denote the width of E (that is, the length of (E)) so that EEw(E) is the perimeter of Π; let EκE be the w(E) × κ rectangle with side (E). Finally, let H(E) be the open half plane that contains E and whose boundary contains (E); then, for a given E, the set of x ∈ Π with R(x) = H(E) is a proper subset of Eκ. Some aspects of these definitions are illustrated in Fig. 4.

FIG. 4.

Constructions related to the edge term. On the left, an example of a polygonal region Π with one of the half-open line segments of the boundary highlighted. On the right, (part of) the corresponding semi-infinite rectangle E and half plane H(E); note EκEH(E).

FIG. 4.

Constructions related to the edge term. On the left, an example of a polygonal region Π with one of the half-open line segments of the boundary highlighted. On the right, (part of) the corresponding semi-infinite rectangle E and half plane H(E); note EκEH(E).

Close modal
Letting x denote the component of x perpendicular to (E), we have
and thus,
with a1 as defined after Eq. (19). In addition,

Finally, let C be the set of CR2, which are open infinite wedges matching the corners of Π. For such a C, let φ(C) be its opening angle, and let E1(C), E2(C) be the elements of E associated with the line segments incident on the corner; thus, C = H(E1(C)) ∩ H(E2(C)) if φ(C) ≤ π and C = H(E1(C)) ∪ H(E2(C)) if φ(C) > π (Fig. 5). In addition, let v(C) be the vertex of C, and let Cκ be the set of xC, which are within a distance κ of both edges of C or, equivalently, the set of x ∈ Π for which RΠ(x) = C. For j = 1, 2, let Ĉκj=(Ej)κ(C)\C, and let Ĉj be the corresponding infinite wedge; both of these are ∅ if φ(C) ≥ π/2.

FIG. 5.

Two examples of the notation for the definitions of the corner term; Cκ is drawn in dark gray and a portion of Eκ1,Eκ2 in light gray (cf. Fig. 3). Note that in the example on the left, Eκ1,Eκ2 extend outside of C (and outside of Π); these extending portions are Cκ1,Cκ2.

FIG. 5.

Two examples of the notation for the definitions of the corner term; Cκ is drawn in dark gray and a portion of Eκ1,Eκ2 in light gray (cf. Fig. 3). Note that in the example on the left, Eκ1,Eκ2 extend outside of C (and outside of Π); these extending portions are Cκ1,Cκ2.

Close modal
The point of all this is
(note that the integrals over Ĉκj cancel the part of the Eκ integrals where x ∉ Π), where
where Êj is the open quarter-plane with vertex v(C) obtained by extending Eκj. PC can be bounded using Theorem 3, and it satisfies |PC(x;t)|2C3t1ek|xv(C)|2/t for k=k(Π)=18minCCsin2φ(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 E2 than E1, see Fig. 6), I write
and as can be seen in Fig. 6, the distances involved in the las two differences are of the desired order. From Eq. (47), we see that the same is true of ∂P(xt) for xĈ1Ĉ2. Consequently, we see that the integrals in
are convergent; by rescaling 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
combining this with Eqs. (46) and (49) and letting a2(Π)=CCa2(C), we get
we obtain Eq. (19).
FIG. 6.

Example of the estimates on PC(x;t) in the case φ(C) ≤ π/2; here, PC=PC+PR2PH(E1(C))PH(E2(C)). For the indicated x, y is both the closest point in CH(E1(C)) and the closest point in H(E2(C))R2 and |xy|sin(φ(C)/2)xv(C).

FIG. 6.

Example of the estimates on PC(x;t) in the case φ(C) ≤ π/2; here, PC=PC+PR2PH(E1(C))PH(E2(C)). For the indicated x, y is both the closest point in CH(E1(C)) and the closest point in H(E2(C))R2 and |xy|sin(φ(C)/2)xv(C).

Close modal
Let Bs be a two dimensional Brownian bridge, shifted and rescaled, so that B0=Bt=x; we then have
where DΩ(B) is the indicator function of the event that BsΩ for all s ∈ [0, t]; recall P0(t)=PR2(x,x;t)=1/2πt. Similarly, letting B̃m be the process obtained by conditioning the random walk W̃m to return to its starting point after 2n steps,
where P̃0(n)=P̃R2(x,x;n) is the probability that a simple random walk on Z2 returns to its starting point after n steps; representing this in terms of two independent 1D walks as in Eq. (45),
for large n using Stirling’s formula.
Combining Eqs. (58) and (59) gives
note that the endpoints 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.

Theorem 5.
There is a constant C4 > 0 such that for any yZ2, xR2, with |xy| ≤ 1/2 and any integer n > 1, there exists a coupling of Bs and B̃s 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 B̃ 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 |xy| ≤ 1/2 ≤ log n.

Using this representation,
For any 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 B̃ come within a distance δ of Ωc but at least one of them remains entirely in Ω. This, in turn, requires that in the same time interval, Bs leaves
but does not leave
Using this observation with δ = C4 log n and noting that |12P0(n)/P̃0(2n)|C5/n [recall Eq. (60)], Eq. (61) gives
The first term on the right-hand side appears to be difficult to estimate precisely when Ω is not convex and when the distance of 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.

Lemma 6.
For all τ > 0, ΩR2, and all xR2,
with C7 = 2π/e.

Using the Markov property of Brownian motion,
for any positive integer n; as long as τe|Ω|/2π, we can always choose
to obtain Eq. (67), and for smaller 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:

Definition 7.

An S(ρ) set is an open set ΩR2 such that any point yR2\Ω is an endpoint of a line segment y of length ρ, which does not intersect Ω.

Note that the polygon Π we consider is an S(1) set and LΠ is an S(L) set; Π+(δ) and Π(δ) [respectively, (LΠ)+(δ) and (LΠ)(δ) are S(1/2) (respectively, S(L/2)] for δ < C8 (respectively, δ/L < C8) for some Π-dependent C8.

Lemma 8.
There exists C9 > 0 such that if τ > 0, Ω is an S(τ) set, and x ∈ Ω, then
with R being the distance between x and Ω, as long as Re2/eτ.

Let y be a point in Ω with |xy| = R, and let y be the line segment of length τ associated with y by Definition 7. Since Ωc, trivially
for any ρ > 0. For ρτ, 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 and y,
and using this along with Lemma 6, we obtain
Choosing ρ=2e1/2τ/(logτ2logR) (which satisfies ρτ under the assumption Re2/eτ), this gives Eq. (70).□

This can then be use to obtain an estimate on the heat kernel, which will be useful for estimating P(LΠ)+(δ), and hence the difference, near the boundary where any Brownian motion is very likely to exit (LΠ)+(δ):

Lemma 9.
There exists C10 > 0 such that if t > 0, Ω is an S(τ) set for some τt/3 and x ∈ Ω, then
with R being the distance between x and Ω, as long as Re2/eτ.

Noting that
using the assumption τt/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 (LΠ)(δ), manages to travel far enough to get back to its starting point without leaving (LΠ)+(δ), whose boundary is now at a distance of order δ.

Lemma 10.
There exists C11 for which the following holds. For any S(ρ), set Ξ, any open measurable Ω ⊂ Ξ with maxyΩ minzΞ|yz| = δ, and any x ∈ Ω with dist(x, Ω) = R,

First, using the strong Markov property and conditioning on the joint distribution of TΩ and WTΩ,
similarly, conditioning on the event that Ws leaves a certain disk Bq(z), which contains y but not x before exiting Ξ,
Letting z be a point on the boundary of Ξ with |zy| ≤ δ′, for all qρ, we can use the assumption that Ξ is an S(ρ) set as in Eq. (72) to bound PyTBq(z)TΞ(4/π)q/δ, and noting P(w, xu) ≤ 1/(|wx|2) ≤ 1/((Rq)2), Eq. (76) follows by choosing q = min(ρ, R/2).□

Lemma 11.
There exist C12, C13, C14, C15 > 0, depending on Π, such that for δC12L and dist(x, Π+(δ)) = R,
whenever C14t34L2, and
when 34L2tC15L5/2.


First of all, there exist C12, C16 < such that for all δC12, Π+(δ) is an S(1/2) set and maxyΠ(δ)minzΠ+(δ)|yz|C16δ; rescaling, this implies that (LΠ)+(δ) is S(L/2) and maxy(LΠ)(δ)minz(LΠ)+(δ)|yz|C16δ for δC12L.

Then, by choosing C14 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 C15 small enough likewise for the first case in Eq. (80) with τ = L2/4. The other cases follow from Lemma 10 with ρ = L/2.□

Integrating in x, we obtain the following corollary:

Corollary 12.
There exists C17 such that
whenever δC12L and C14tC15L5/2.

We now arrive at the conclusion of this section:

Theorem 13.
For each Π, there exists C18 < ∞ such that whenever KN and e9/2KL,

Using Eq. (66), we have
with δ = C4 log n. For nC15L5/2, we can bound the first integral using Corollary 12 and the second using P(LΠ)+(δ)(x,x;n)1/4πn, giving
For n > C15L5/2, 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.□

In this section, I control the difference between the discrete heat kernel on Π and in the reference geometry and relate the latter to the terms α0, α1, α2 in Eq. (5). To take into account the presence of a lattice scale, we define
We now introduce counterparts of a0(Π), a1(Π), a2(Π); however, their time dependence is not so simple as in the continuum case. For H a half-plane, we let
[cf. Eq. (47)], and define P̃C by the analog of Eq. (52). We then have
To extract the asymptotics we are interested in, let
Note that A2 is actually independent of L, A2(LΠ, n) = A2(Π, n), since rescaling each wedge CC by an integer factor gives the same result; it is also invariant under the symmetries of Z2, and in fact, Â2(C,n) is evidently determined by the opening angle of C and its orientation relative to the lattice. Analogously, A1(LΠ, n) = LA1(Π, n), and Â1(E,L,n) is fixed by the length and orientation of the edge of LΠ corresponding to E.
Subtracting these terms from Eq. (87), we can repeat the analysis of Sec. IV to write the remainder in terms of differences between (discrete) heat kernels on domains, which differ at distance at least κL from the points of interest, which can be bounded using Theorem 4, giving the counterpart of Eq. (56),
for nL2. Applying Theorem 4 again to bound the difference between P̃LΠR and P̃LΠ gives an estimate of similar magnitude,
and consequently,
for all 1 < KL.
Let us now compare these contributions to their continuum counterparts. For A0, this is quite simple: recalling Eq. (46),
using Pick’s formula to estimate the difference in areas.
For A1, recalling Eq. (49), we have
This difference can be bounded using a similar approach to Sec. V, with a few variations and added elements. To begin with, note that
[cf. Eq. (61)]. The difference in probabilities on the right-hand side of Eq. (96) is bounded by the probability that exactly one of the bridge processes leaves the half plane H(E); as in Eq. (66), we can bound this difference using Theorem 5, giving
for all n > 1, |xy| ≤ 1/2, with δ = C4 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
(a better bound can, of course, be obtained from the exact expression for the Dirichlet heat kernel on the half-plane, but it would not change the final result). This is not enough by itself to bound the summand in Eq. (95), since Eq. (97) also contains a term of order 1/n2, which does not depend on the distance from the boundary; however, it does give
where ER1 is a rectangular region like Eκ (cf. Fig. 4), with R1 being any number such that |ER1|=#ER1Z2. Further away from the edge, we can use Theorems 3 and 4 to bound ∂PH(E) and P̃H(E) separately, and setting R1 as close as possible to n19/14, the contribution from E\ER1 is superexponentially small for large n. In conclusion, we have
for n > 1; note, in particular, that this decays faster than 1/n for n large.
For the corner contributions, we can apply the same techniques to bound PC(x;n)12P̃C(y;2n), giving estimated in terms of distances to the vertex v(C), by the same reasoning as in the derivation of Eq. (55); so with R1 = n25/28, we have
for n large enough.
I now define
for e an edge of LΠ, Ee the associated element of E(LΠ), c a corner, and Cc the associated wedge in C(LΠ). 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,
Note that Eqs. (94), (100) and (101) imply that the above sum converges; the last term in the summand formally gives the divergent integral a2(Π)1dtt whose divergence is canceled by A2.
Using Eqs. (11) and (17),
Recalling Eq. (93),
for L/K, while in Theorem 13, we saw that
for K, L/K, K9/7/L. All of the error terms will ultimately have similar forms, so I will set K=ceil(C34L/logL) with C34 chosen large enough so that the exponential error terms are negligible; we then have
for L.
Recalling the definition of FΠ in Eq. (20) and the relationship between PΠ,Σ and Tr etΔΠ,Σ (Theorem 2),
which, combined with Eqs. (94) and (100), yields
with K chosen as discussed above. Rescaling F as in Eq. (27) and taking advantage of the small t asymptotics in Eq. (19), we have
and combining this with Eqs. (109) and (111) gives
Finally, using Eq. (101) to bound the error made in extending the last sum to infinity and recognizing α0, α1, and α2 using Eq. (105) and α4(LΠ,LΣ)=ζΔLΠ,LΣ(0) using Eq. (26), we obtain Eq. (8). As mentioned before, Theorem 1 then follows by using Eq. (28).

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.

The author has no conflicts to disclose.

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

Let ΠΣ denote a double cover of Π branched around each point in Σ, and let p: ΠΣ → Π be the associated projection map. Let L2(Π, Σ) be the set of equivalence classes (with respect to almost-everywhere equality) of square-integrable functions f:ΠΣR with the property that f(x) = −f(y) if p(x) = p(y) but xy. L2(Π, Σ) is evidently a closed linear subspace of L2Σ) and so is itself a Hilbert space with the same inner product. Let C0(Π,Σ) be the set of smooth functions fL2(Π, Σ) such that for all yΠΣ, limxyf(x) = 0 and the corresponding limits of all of the derivatives of f exist.

The Laplacian ΔΠ,Σ is defined as the Friedrichs extension of the (positive) Laplacian −∇ · ∇ on C0, which is self-adjoint with respect to the inner product of L2(Π, Σ). It is a strictly positive operator: letting ΔΠ be the Dirichlet Laplacian on Π (not the double cover), for any fC0(Π,Σ), the function f̃ defined by f̃(p(x))=|f(x)| is in the domain of ΔΠ; we can extend it to a function in the domain of ΔR for R a rectangle containing Π, and then,
where in each expression, (⋅, ⋅) denotes the appropriate L2 inner product; of course, the spectrum of ΔR 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 ΔΠ,Σ. Among other things, this combines with the spectral theorem to imply that etΔΩ,Σ is defined for all t ≥ 0.
For W a Brownian trajectory on R2, which does not intersect Σ (which is finite, so these trajectories form a full measure set), let WΣ denote its continuous lift to the dual cover of R2 branched around Σ, starting from a specified starting point x ∈ ΠΣ. Consider the family of operators Kt on L2(Π, Σ) defined by
For fC0(Π,Σ), by definition f(WTΠΣ)=0, and so fWtΣ1[t,)(TΠ)=f(WtTΠΣ); bearing in mind that ΔΠ,Σ acts locally on smooth functions like the ordinary Laplacian, Itô’s formula implies that
by continuity, this implies ddtKt=12KtΔΠ,Σ as operators on L2(Π, Σ), and evidently, K0 is the identity; in other words, Kt is a one-parameter group generated by 12ΔΠ,Σ; there is only one such group (see Ref. 34, Theorem 1.7), so Kt=e(t/2)ΔΠ,Σ.

Theorem 14.
For any closed, simply connected B ⊂ Π, there is a function QB(x, yt) from ΠΣ × p−1(B) × (0, ) to R such that
for any gL2(Π, Σ) with supp gB; furthermore, QB is C as a function of y, and
where P is the full-plane heat kernel introduced in Eq. (33).

Let B1, B2 ⊂ Π be open and simply connected with B̄1BB2 and δd(B, ∂B1) ∨ d(B1, ∂B2) > 0. Letting W denote a planar Brownian motion started at x, I introduce two related sequences of hitting times as follows: let τ0 = σ0 = 0, and for j = 1, 2, …,
with these definitions (thanks, in particular, to the nonzero distance between B1 and the boundary of B2), J≔ max{j = 0, 1, …: σjt} is almost surely finite. Furthermore,
and so, recalling Eq. (A2),
using the strong Markov property, where
Since B2 is simply connected, thanks to the indicator function, G(ys) is an integral over paths, which remain in the same connected component of p−1(B2) as y=WσJΣ. To spell this out, I introduce the following notation: Let B21 and B22 be the two connected components of B2,
and gκ:B2R, xg[p|B2κ]1(x). Then,
where PB2 is the Dirichlet heat kernel as introduced in Eq. (34). Returning to Eq. (A9), we then have
The bounds in Eq. (A5) are evident from the first form of this definition, since PB2 and 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 σJ = 0) or y∂B1∂B2 [and thus |yp(z)|≥ δ]. With the other arguments so restricted, P is uniformly smooth as a function of z, and so we also see that QB is smooth as claimed.□

Given that they are smooth in the second argument, any two kernels QB(x, ⋅; y) and QB(x, ⋅; t) defined on different domains coincide on the intersection of their domains, and so we can extend by linearity to the rest of L2(Π, Σ) and obtain the following corollary:

Corollary 15.
There is a function QΠ,Σ:ΠΣ×ΠΣ×(0,)R such that
for any fL2(Π, Σ); furthermore, QΠ,Σ is C as a function of y, and

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

Proof of Theorem 2.
For any simply-connected, Borel-measurable U ⊂ Π, let fU:ΠΣR 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, fUL2(Π, Σ), (fU, fU) = 2|U|, and
γU(x,y)=fU(x)fU(y)=1,x and y are in the same connected component of p1(U),1,otherwise,
which is independent of which fU was chosen. Note also that if diam U is small enough, in the last integral, γ(W0Σ,WtΣ) is equivalent to eiπWΣ(W;t) (recall that WΣ was defined in the statement of Theorem 2 to take integer values). Examining Eq. (37) in light of this, we see that for x ∈ ΠΣ, the definition of PΠ,Σ there gives
thanks to the smoothness of QΠ,Σ shown in Corollary 15.
Let UN, for N a positive integer, denote the nonempty sets that coincide with a connected component of a set of the form Π2N[m,m+1)×[n,n+1) for some m,nZ; 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
the sequence fU11,,fU1|U1|,fU21, is an overcomplete basis of L2(Π, Σ), and by applying the Hilbert–Schmidt process to this sequence, we obtain an orthonormal basis ψ1, ψ2, … such that
for all positive integer N. Since etΔΠ,Σ is non-negative, this suffices to show that
Tr etΔΠ,Σ=j=1(ψj,etΔΠ,Σψj)=limNUUN(fU,etΔΠ,ΣfU)(fU,fU).
Using Eq. (A17),
where UN(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 2t, i.e., to PΠ,Σ(p(x), p(x); 2t), and this together with the bound in Eq. (A16) provide the conditions to apply the dominated convergence theorem and obtain
Tr etΔΠ,Σ=12ΠΣPΠ,Σ(p(x),p(x);2t)dx,
which, passing to an integral over x ∈ Π, is equivalent to Eq. (36).□

Lemma 16.
For all integers n ≥ 1 and 0 ≤ mn,

First, note that if m = 0 or m = n, the bound is obvious, so we consider 1 ≤ nn − 1. In this case, using Stirling’s formula with explicit error estimates,35 
Letting Ln(μ) = (μ + 1/2) log  μ + (nμ + 1/2) log(nμ), we have Ln(n/2)=0,
evidently, n/2 is the unique local minimum of Ln(μ). It is easy to see that
for n large enough (in fact, for n ≥ 4), and so we have
Noting that the last term is bounded by 1/2, this combines with Eq. (B2) to give
for sufficiently large n, and the result can be trivially extended to smaller n.□

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