Discrete and zeta-regularized determinants of the Laplacian on polygonal domains with Dirichlet boundary conditions
aa r X i v : . [ m a t h - ph ] F e b Discrete and zeta-regularized determinants ofthe Laplacian on polygonal domains withDirichlet boundary conditions
Rafael L. Greenblatt ∗ SISSA, Trieste, ItalyFebruary 10, 2021
Abstract
For Π ⊂ R a connected, open, bounded set whose boundary is afinite union of polygons whose vertices have integer coordinates, thelogarithm of the discrete Laplacian on L Π ∩ Z with Dirichlet boundaryconditions has an asymptotic expansion for large L in which the termof order 1 is the logarithm of the zeta-regularized determinant of thecorresponding continuum Laplacian.When Π is not simply connected, this result extends to Laplaciansacting on two-valued functions with a specified monodromy class. For a domain Ω ⊂ R , let G (Ω) be the graph whose vertex set is e Ω := Ω ∩ Z and whose edge set E (Ω) is the set of pairs { x, y } ⊂ e Ω such that the linesegment xy has length one and is entirely contained in Ω (see Figure 1 forexamples). The discrete Laplacian on Ω with Dirichlet boundary conditionsis the operator e ∆ Ω on ℓ (Ω ∩ Z ) given by e ∆ Ω f ( x ) = 4 f ( x ) − X y ∈ Z xy ⊂ Ω | x − y | =1 f ( y ) , (1) ∗ e-mail address: [email protected] xy is the closed line segment from x to y . Note that for bounded Ω , this is a symmetric matrix with eigenvalues in (0 , . The main focusof this manuscript is studying det e ∆ Ω of the following form: we will fix abounded open set Π ⊂ R whose boundary is a disjoint union of polygonswhose vertices are all in Z , and consider Ω = L Π for integer L . ThroughoutFigure 1: Example of a shape Π and the graphs G ( L Π) for different integervalues of L (not to scale).this work, I will use Ω for a general domain and Π for a bounded domain ofthis polygonal form.In fact I will consider a somewhat more general Laplacian. For an as-signment of ρ xy ∈ C to each x, y ∈ Ω ∩ Z with | x − y | = 1 such that ρ xy = 1 /ρ yx , we can define the discrete scalar Laplacian on Ω with Dirichletboundary conditions and connection ρ as e ∆ Ω ,ρ f ( x ) = 4 f ( x ) − X y ∈ Z xy ⊂ Ω | x − y | =1 ρ xy f ( y ) . (2)Writing out det e ∆ Ω ,ρ by the Leibniz formula, ρ appears in the form of prod-2cts n Y j =1 ρ y j ,y j +1 (3)where y , . . . , y n +1 is a sequence which has no repeated elements except y n +1 ≡ y ; these sequences can be naturally identified with simple closedcurves; I will call them monodromy factors. I will consider connections withthe following property: for some finite Σ ⊂ Ω c , the monodromy factor is − if the associated closed curve winds around an odd number of elements of Σ and otherwise. From the above considerations, det e ∆ Ω ,ρ is the same for all ρ with this property for the same Σ , so we can let det Σ e ∆ Ω := det e ∆ Ω ,ρ (4)for an arbitrary chosen such connection; it is easy to construct such a ρ forany Σ . In fact in this way we obtain all of the connections which are “flat”(that is, for which the monodromy factor is for all contractible curves) andtake values only in ± .These determinants are the partition functions of certain sets of essentialcycle-rooted spanning forests [Ken11]. In the simply connected case (where Σ plays no role) the set of forests in question is simply the set of spanning treeson the graph formed by adding a “giant” vertex to G (Ω) which is connectedto all the vertices on the boundary, and this statement is a restatement ofthe Kirchoff matrix-tree theorem. In this case, and also for the case when Ω is genus 1 and Σ consists of a single point in the finite component of Ω c ,this partition function is in turn equal to the number of perfect matchings(configurations of the dimer model) on a “Temperleyan” graph constructedfrom G (Ω) ; however for higher-genus planar domains this correspondenceinvolves a set of forests which has a different characterization [BLR19].The main result of the present paper is the following: Theorem 1. log det L Σ e ∆ L Π = (cid:0) ( L Π) ∩ Z (cid:1) α + X x ∈ ∂ ext ( L Π ∩ Z ) α ( B ( x ) ∩ L Π)+ α (Π) log L + α (Π , Σ) + O (log L ) L / ! (5) as L → ∞ , where1. denotes the cardinality; σ Figure 2: Example of a region Π and set Σ = { σ , σ } used to specify mon-odromy factors; in this example the solid curves have monodromy 1 and thedashed curves have monodromy − , independent of their orientation. ∂ ext denotes the exterior boundary, that is the set of points in Z \ L Π with at least on neighbor in L Π ; B ( x ) is the closed disk of radius centered at x ;3. α (Π) is the sum over corners of Π of ( π − θ ) / πθ where θ is theangle measured in the interior of the corner;4. α (Π , Σ) is the logarithm of the ζ -regularized determinant of the con-tinuum Laplacian on Π with a connection compatible with Σ , definedin Section 2 below. 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) partitionfunction for free quantum field theories and has a very accessible relationshipto the geometry of the domain on which it is defined; it has consequentlybeen used to study the Casimir effect [DC76], the bahavior of quantum fieldtheory in curved space-time [Haw77], and conformal field theory [Gaw99].When Π is simply connected (i.e. a polygon) has been described in greatdetail in [AS94]. Among other things, the fact that α is the logarithm of4uch a determinant implies that α ( L Π , L Σ) = α (Π) log L + α (Π , Σ) (seeEquations (23) and (121)), so Equation (5) is equivalent to log det L Σ e ∆ L Π = (cid:0) ( L Π) ∩ Z (cid:1) α + X x ∈ ∂ ext ( L Π ∩ Z ) α ( B ( x ) ∩ L Π)+ α ( L Π , L Σ) + O (log L ) L / ! , (6)which we in fact prove first.One consequence of Equation (5) is that, for two different Σ on the same Π , det L Σ e ∆ L Π det L Σ e ∆ L Π = exp (cid:18) α (Π , Σ ) α (Π , Σ ) (cid:19) + o (1) , L → ∞ . (7)Expansions to this order for determinants of Laplacians (or dimer parti-tion functions, or Ising model partition functions, which are closely relatedunder suitable conditions) have now been studied for at least two genera-tions. The presence of a term of constant order is interesting for applicationsto statistical physics, for example since it is related to the notion of centralcharge in conformal field theory [BCN86; Aff86].When comparing these results it is important to note that boundary con-ditions (which, in the dimer model, are expressed in details of the geometrychosen) can have dramatic effects. For example, the first publication contain-ing asymptotic formulas of this type [Fer67] considered the partition functionof the dimer model on a discrete rectangle, which is not a Temperleyan graph,and so does not correspond to Dirichlet boundary conditions; consequentlythe expansion obtained there (which, in particular, has no logarithmic term)is not a special case of Theorem 1, although it has some similarities.The aforementioned first publication, by Ferdinand in 1967, studied thediscrete torus as well as a rectangle, and noted the presence of a term of con-stant order, expressed in terms of special functions. This expansion startedfrom the expression of the partition function as a Pfaffian (square root of thedeterminant) or linear combination of Pfaffians of matrices which had beenexplicitly diagonalized, and then carried out an asymptotic expansion usingthe Euler-Maclaurin formula. Subsequently this approach was extended toother geometries, such as the cylinder and Klein bottle, and to obtain theasymptotic series to all orders in the domain size [IOH03; BEP18].The first work to consider a case included in Theorem 1 was [DD83, inparticular Equation (4.23)], who examined the determinant of the Laplacianon a rectangular lattice with Dirichlet boundary conditions (via the product5f the nonzero eigenvalues of the Laplacian with Neuman boundary condi-tions on the dual lattice, adapting an earlier calculation [Bar70], which wasthe first to include a logarithmic term); this was the first work to explicitlyidentify part of the expansion of the discrete Laplacian determinant witha zeta-regularized determinant. Subsequently, Kenyon [Ken00] used the re-sult of this calculation (which was based on an exact diagonalization of theLaplacian), combined with a formula using dimer correlation functions torelate partition functions on different domains, to obtain a result for rec-tilinear polygons, that is, simply connected domains with sides parallel tothe axes of Z . Kenyon gives an alternative characterization of the term α ,related to the limiting average height profile of the associated dimer model;recently Finski [Fin20] showed that this is in fact the logarithm of the zeta-regularized determinant in a work which also generalized the result to, amongother things, nonsimply-connected domains (including nontrivial vector bun-dles), albeit still with the restriction that the boundary of the domain shouldalways be parallel to one of the coordinate axes.In the subsequent two decades there have also been a number of worksrelating the determinant of the Laplacian on higher-dimensional analogs ofthe discrete torus or rectangle [CJK10; Ver18; HK20] to the correspond-ing zeta-regularized determinants. Although these authors introduced ideaswhich illuminate several aspects of the problem and make it possible to treatarbitrary dimension, they only treated graphs for which the spectrum of theLaplacian is given quite explicitly (although [CJK10] does not use this di-rectly, their treatment is based on the fact that the discrete torus is a Cayleygraph, which is essentially the reason its spectrum can be calculated so ex-plicitly). For some self-similar fractals, the problem has been approachedusing recursive relationships which characterize the spectrum less explicitly[CTT18], but this leads us into a rather different context.As in [CJK10], the proof of Theorem 1 is based on the relationship be-tween the (matrix or zeta-regularized) determinant to the trace of the heatkernel, reviewed in Section 2; however, once this relationship is introduced, Itreat the heat kernel using only a probabilistic representation, as the transi-tion probability of a random walk or Brownian motion. In Section 3, I intro-duce this representation (which is nonstandard only because of the presenceof nontrivial monodromy factors) and present some straightforward boundson the effect of changes in the domain Ω and the set Σ which encodes themonodromy. Such bounds can be used to obtain an expansion for the be-haviour of the trace of the continuum heat kernel at small time (small, that is,on the scale associated with the domain); in Section 4, I review the proof ofthis expansion givne in [Kac66], which also provides the occasion to introduce6ertain definitions which will be used again later on to identify correspondingcontributions in the discrete heat kernel. For times which are large on thelattice scale, the trace of the discrete heat kernel converges to that of the cor-responding continuum heat kernel; in 5 I give quantitative estimates on thisconvergence, based on the dyadic approximation. Then in Section 6 I use acombination of the methods of the previous sections to control the behaviorof the lattice heat kernel in a regime corresponding to that of the expansionfor its continuum counterpart considered in Section 4. Finally, in Section 7I combine all of these estimates to conclude the proof of Theorem 1.Theorem 1 is limited to (1) planar regions, (2) polygons whose cornershave integer coordinates, (3) Dirichlet boundary conditions, and (4) subsetsof the square lattice, rather than more general planar graphs; let me commenton these limitations.1. There does not seem to be any particular difficulty in extending theresult to higher-dimensional polytopes, should such a result be of in-terest; however several of the motivations for studying the problem areparticular to the two-dimensional case. Similarly, it is quite straightfor-ward to generalize this treatment to a torus, cylinder, Klein bottle, orMöbius strip, but these cases are all already well understood by othermeans [Fin20]. Domains with punctures or slits [KW20] may also beaccessible with some technical improvements.2. The restriction to this class of polygonal domains is related to the con-struction of the term involving the “surface tension” α in Equation (5).Near the edges of the domain and away from the corners (the situationat the corners is more complicated, but in the end the same considera-tions come into play), the domain coincides with an infinite half-planein regions of size proportional to L , so that the heat kernel there is verywell approximated by the heat kernel on the intersection of Z with thathalf plane. The construction of α , given in Section 6, takes advantageof this, using the fact that, because the slope of the boundary of thehalf-plane is rational, the result is a periodic graph.3. The discrete heat kernel in Dirichlet boundary conditions has a particu-larly convenient probabilistic representation (see Section 3 for details):for a trivial connection, the diagonal elements are simply the probabil-ities that a standard continuous-time random walk on Z returns to itsstarting point at a specified time without ever having left the domainThis characterization has many helpful properties (most immediately,it is strictly monotone in the choice of domain). For a general connec-tion, it is the expectation value of a function which is the monodromy7actor of the trajectory of the random walk if it returns to its startingpoint without leaving the domain and zero otherwise, which is obvi-ously bounded in absolute value by the probability used in the formercase. The continuum heat kernel has a similar representation in termsof standard two-dimensional Brownian motion.The probabilistic represerntation for the Laplacian with Neumann bound-ary conditions is less immediately tractable; in two dimensions thismight seem to be a moot point since there is a duality between Neu-mann and Dirichlet boundary conditions, but it the problem returns ifone wishes to treat mixed boundary conditions.4. The restriction to the square lattice is most crucial for the techniqueI use to obtain quantitative control on the approximation between thediscrete and continuum heat kernels in Section 5. I use an approachbased on the dyadic coupling, which is essentially a one-dimensionaltechnique, but which can be applied to continuous-time random walkson Z d and a few other graphs where the coordinates of the walker evolveindependently.There are a number of interesting problems which could be studied ifthis limitation could be overcome. In the last few years, similar expan-sions for periodic graphs imbedded in the torus [KSW16] or Klein bottle[Cim20] based on the characteristic polynomial method have found aconstant term which is universal (i.e. independent of many of the de-tails of exactly which graph is chosen); it seems reasonable to expectthat the same is true for some class of planar graphs. Furthermore,the determinants of Laplacians of a large class of graphs embedded innonplanar manifolds appear in the study of Liouville quantum gravityas partion functions of the discrete Gaussian free field and other latticemodels [Ang+20]. Acknowledgements
I thank Bernard Duplantier for introducing me to this problem, and pointingout its relationship to the heat kernel. The outline of the proof occurred tome during the workshop “Dimers, Ising Model, and their Interactions” atthe Banff International Research Station, and I thank the organizers for theinvitation to attend the workshop as well as the many participants in thediscussions which provoked it. I would also like to thank Alessandro Giulianifor discussing the work with me through all the stages of development.This work was conducted at the Università degli Studi Roma Tre (Rome,Italy) with the support of the European Research Council (ERC) under the8uropean Union’s Horizon 2020 research and innovation programme (ERCCoG UniCoSM, grant agreement n.724939).
For M a symmetric m × m matrix with positive eigenvalues < µ ≤ µ ≤ . . . , µ m , we define an entire function by ζ M ( s ) := m X j =1 µ − sj ; (8)we then have ζ ′ M (0) = − m X j =1 log µ j = − log det M (9)or equivalently det M = exp ( − ζ ′ M (0)) ; to evaluate this, we note that for Re s > ζ M ( s ) = m X j =1 s ) Z ∞ t s − e − tµ j d t = 1Γ( s ) Z ∞ t s − Tr e − tM d t (10)where Γ( s ) := R ∞ t s − e − t d t , and consequently ζ ′ M ( s ) = 1Γ( s ) Z ∞ t s − log t Tr e − tM d t − ψ ( s )Γ( s ) Z ∞ t s − Tr e − tM d t, (11)where ψ ( s ) := Γ ′ ( s ) / Γ( s ) . Using obvious bounds for Tr e − tM and well-knownasymptotics for ψ and Γ near the origin [DLMF, eq. 5.7.1, 5.7.6] we canobtain ζ ′ M (0) = lim s → + ζ ′ M ( s ) = Z e − γ Tr (cid:0) e − tM − I (cid:1) d tt + Z ∞ e − γ Tr e − tM d tt (12)In our case, we are therefore interested in Tr e − t e ∆ Ω ,ρ . Writing out theexponential as a power series and expanding the trace and matrix powers,once again ρ appears only through the monodromy factors, so we will write Tr Σ e − t e ∆ Ω = Tr e − t e ∆ Ω ,ρ for any ρ associated with Σ .To obtain comparable continuum entities, let L (Π , Σ) denote a smoothreal line bundle over Π with a connection such that parralel transport arounda closed curve corresponds to multiplcation by − if the curve winds around9n odd number of points from Σ and 1 otherwise, and let ∆ Π , Σ be the Lapla-cian on L (Π , Σ) with Dirichlet boundary conditions . For consistency withEquation (1) we use the positive Laplacian, giving a sign change with respectto many of the references cited.The spectrum of ∆ Π , Σ is discrete and bounded from below, and its densityof states is asymptotically approximately linear (Weyl’s law), so the zetafunction ζ ∆ Π , Σ ( s ) := X λ ∈ spec ∆ Π , Σ λ − s (13)is well defined and analytic for Re s > ; as we shall shortly see, ζ ∆ Π , Σ can be analytically continued to a neighborhood of 0, and – by analogywith Equation (9) – e − ζ ′ ∆Π , Σ (0) is called the ζ -regularized determinant of theLaplacian.The analogues of Equations (10) and (11) holds for Re s > (using Weyl’slaw to show that e − t ∆ Π , Σ is trace class for t > ); its trace has has a note-worthy asymptotic expansion [BB62] Tr e − t ∆ Π , Σ = a (Π) t − + a (Π) t − / + a (Π) + O ( e − δ (Π) /t ) , t → (14)where a (Π) is / π times the area of Π , a (Π) is − / √ π times the lengthof the boundary, and a (Π) is the sum over corners of ( π − θ ) / πθ where θ is the opening angle measured from the interior of Π ; an expansion forsmooth manifolds with smooth boundaries (or no boundaries) is also known[MS67; Gil94]. Although [BB62; Kac66; MS67] were formulated for the planeLaplacian or Laplace-Beltrami operator (equivalent to Σ = ∅ ), the proof ofEquation (14) in [Kac66] extends straightforwardly to the other connectionswe consider. I will present a version of this proof in Section 4, both forcompleteness and because some of the entities introduced there are helpfulfor comparison with the discrete case. Tr e − t ∆ Π , Σ also decreases exponentially for sufficiently large t , since thespectrum of ∆ Π , Σ is bounded away from 0, and so letting F Π , Σ ( t ) := Tr e − t ∆ Π , Σ − a (Π) t − − a (Π) t − / − a (Π) [0 ,e − γ ] ( t ) (15) This is defined as usual as the continuous extension of the Laplacian acting on smoothsections of L (Π , Σ) which vanish at the boundary of Π . The first formula for a is in [Kac66]; this simpler version, due to D. B. Ray, was firstpublished in [MS67].
10e obtain, for any
K > e − γ and Re s > , Z K t s − Tr e − t ∆ Π , Σ d t − Z K t s − F Π , Σ ( t ) d t = Z K (cid:0) a (Π) t s − + a (Π) t s − / (cid:1) d t + a (Π) Z e − γ t s − d t = a (Π) K s − − s + a (Π) K s − / − s + a (Π) s e − γs ; (16)we thus have Γ( s ) ζ ∆ Π , Σ ( s ) = Z ∞ K t s − Tr e − t ∆ Π , Σ d t + Z K t s − F Π , Σ ( t ) d t + a (Π) K s − − s + a (Π) K s − / − s + a (Π) s e − γs , (17)and we can analytically continue and then take the limit K → ∞ to obtain ζ ∆ Π , Σ ( s ) = 1Γ( s ) (cid:20)Z ∞ t s − F Π , Σ ( t ) d t + a (Π) s e − γs (cid:21) (18)for all | s | < , and noting thatdd s Z ∞ t s − F Π , Σ ( t ) d t (cid:12)(cid:12)(cid:12)(cid:12) s =0 = Z ∞ F Π , Σ ( t ) log t d tt < ∞ (19)along with lim s → s ) = 0 , lim s → dd s s ) = 1 , lim s → s Γ( s ) = 1 , and lim s → dd s s Γ( s ) = γ (20)gives ζ ′ ∆ Π , Σ (0) = Z ∞ F Π , Σ ( t ) d tt . (21)Note, finally, that from the definitions of F and the various a n (Π) wehave F L Π ( t ) = Tr e − ( t/L )∆ Π , Σ − a (Π) L t − a (Π) Lt / − a (Π) [0 ,e − γ L ] ( t )= F Π , Σ (cid:0) t/L (cid:1) + a (Π) (1 ,L ] ( e γ t ) (22)whence ζ ′ ∆ L Π ,L Σ (0) = ζ ′ ∆ Π , Σ (0) + 2 a (Π) log L. (23)11 Heat kernels, Brownian motion, and randomwalks
I now turn to the represention I will use to estimate the trace of e − t e ∆ L Π , Σ .This is a matrix whose elements are e P D Ω ,ρ ( x, y ; t ) := h e − t e ∆ Ω , Σ i xy = e − t ∞ X n =0 t n n ! h(cid:16) I − e ∆ Ω , Σ (cid:17) n i xy = e − t ∞ X n =0 (4 t ) n n ! X y : | x − y | =1 ρ x,y · · · X y n : | y n − − y n | =1 ρ y n − ,y n δ y n ,y , (24)where I is the identity matrix. When ρ ≡ , this is can be understood as aprobability in terms of a random walk: to be precise, e P D Ω ( x, y ; t ) = P x h f W t = y, e T Ω > t i (25)where f W is a standard continuous-time random walk on Z starting at x ,and e T X , X ⊂ R , denotes the first time at which f W leaves X , which in lightof the definition Equation (1), we understand to be the first time it jumpsalong a bond associated with a line segment not entirely contained in X .Letting W z ( f W ; t ) denote the winding number (index) of f W around z upto time t , when ρ is associated with some Σ , for the diagonal elments (which,ultimately, are the ones we are interested in) this generalizes to e P D Ω , Σ ( x, x ; t ) = E x h { x } (cid:16) f W t (cid:17) ( t, ∞ ) (cid:16) e T Ω (cid:17) e iπW Σ ( f W ; t ) i (26)with W Σ ( f W ; t ) := P σ ∈ Σ W σ ( f W ; t ) .Let us now look at the continuum. With Σ = ∅ , ∆ Ω , Σ is just the usualDirichlet Laplacian ∆ Ω on Ω , and so for t > e − t ∆ Ω is an integral operatorwhose kernel can be written P D Ω ( x, y ; t ) = lim r → + πr P x [ | y − W t | ≤ r, T Ω > t ] , (27)where W is a Brownian motion starting from x , and T Ω is its first exit timefrom Ω .A similar form holds for the more general case. For σ, x ∈ R let ϕ σ ( x ) be the angle between x − σ and the horizontal axis, with values in ( − π, π ] ,and let ϕ Σ ( x ) be P σ ∈ Σ ϕ σ ( x ) + 2 πn ( x ) for whichever integer n ( x ) puts it inthe same interval. Then smooth sections of L (Ω , Σ) are in a natural one-to-one correspondence with functions Ω → R which are smooth except that12 ( x n ) → − f ( x ) when ϕ Σ ( x n ) ց − π modulo π , so that the Laplacian on L (Ω , Σ) coincides with the Laplacian acting on these latter functions whenit is extended by continuity to the points with ϕ Σ ( x ) = π . So e − t ∆ Ω , Σ actingon this space is an integral operator with kernel P D Ω ( x, y ; t ) = lim r → + πr n P x (cid:2) |W t − y | ≤ r, T Ω > t, πW Σ ( W ; t ) + ϕ Σ ( x ) ∈ ( − π, π ] + 4 π Z (cid:3) − P x (cid:2) |W t − y | ≤ r, T Ω > t, πW Σ ( W ; t ) + ϕ Σ ( x ) ∈ ( π, π ] + 4 π Z (cid:3)o . (28)When y = x we can rewrite this as P D Ω , Σ ( x, x ; t ) = lim r → + πr E x (cid:2) [0 ,r ] ( |W t − x | ) ( t, ∞ ) ( T Ω ) e iπW Σ ( W ; t ) (cid:3) , (29)which can be used to calculate Tr e − t ∆ Ω , Σ = Z Ω P D Ω , Σ ( x, x ; t ) d x. (30)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 2.
There is a constant C > such that for all Ω , Θ ⊂ R , Σ ⊂ Ω c ∩ Θ c , x ∈ Ω ∩ Θ , (cid:12)(cid:12) P D Ω , Σ ( x, x ; t ) − P D Θ , Σ ( x, x ; t ) (cid:12)(cid:12) ≤ C t exp (cid:0) − [dist( x, (Θ △ Ω) ∪ (Σ △ Σ ))] / t (cid:1) . (31) Proof.
For brevity let R := dist( x, (Θ △ Ω) ∪ (Σ △ Σ )) . Using Equation (29),we first note that (cid:12)(cid:12) P D Ω , Σ ( x, x ; t ) − P D Θ , Σ ( x, x ; t ) (cid:12)(cid:12) ≤ P ( t ) := 2 P D R , ∅ ( x, x ; t ) = 12 πt , (32)which already gives a bound of the desired form for R < t. In the comple-mentary case, we begin by noting that (cid:12)(cid:12) P D Ω , Σ ( x, x ; t ) − P D Θ , Σ ( x, x ; t ) (cid:12)(cid:12) = lim r ց πr (cid:12)(cid:12) E x (cid:2) [0 ,r ] ( |W t − x | ) (cid:8) ( t, ∞ ) ( T Ω ) e iπW Σ1 ( W ; t ) − ( t, ∞ ) ( T Θ ) e iπW Σ2 ( W ; t ) (cid:9)(cid:3)(cid:12)(cid:12) . (33)13or the difference in the expection to be nonzero, W must either enter Θ △ Ω or wind around a point in Σ △ Σ before time t , either of which requires itto travel a distance at least R from its starting point; thus (cid:12)(cid:12) P D Ω , Σ ( x, x ; t ) − P D Θ , Σ ( x, x ; t ) (cid:12)(cid:12) ≤ r ց πr P x (cid:2) |W t − x | ≤ r, T B R ( x ) ≤ t (cid:3) ≤ P x (cid:2) T B R ( x ) ≤ t (cid:3) sup y / ∈ B R ( x ) s ∈ [0 ,t ] P D R ( y, x ; s ) , (34)where B R ( x ) is the circle of radius R centered at x . The probability of leaving B R ( x ) can be estimated in terms of the probability of moving by more than √ R in any one of the four coordinate directions, and the probability of eachof these events can be calculated using the reflection principle: P x (cid:2) T B R ( x ) ≤ t (cid:3) ≤ √ R √ t ! ≤ √ t √ πR e − R / t (35)where erfc is the complimentary error function, and the last inequality followsfrom the bound erfc( x ) ≤ e − x √ πx , x ≥ . (36)As for the second factor on the right hand side of Equation (34), we have sup y / ∈ B R ( x ) s ∈ [0 ,t ] P D R ( y, x ; s ) = sup s ∈ [0 ,t ] e − R / s πs ≤ eπR , (37)and the claimed result for R > t follows by combining this with Equa-tions (34) and (35).The lattice version is somewhat more complicated. Theorem 3.
There is a constant C > such that for all Ω , Θ ⊂ R , Σ ⊂ Ω c ∩ Θ c , x ∈ Ω ∩ Θ , (cid:12)(cid:12) P D Ω , Σ ( x, x ; t ) − P D Θ , Σ ( x, x ; t ) (cid:12)(cid:12) ≤ C e ρ ( R, t ) (38) with R = dist( x, (Θ △ Ω) ∪ (Σ △ Σ )) and e ρ ( R, t ) := te − R , t ≤ t − e − R , < t ≤ R/ (4 e ) t − exp ( − R / t ) , otherwise . (39)14 roof. Beginning as before, we have (cid:12)(cid:12)(cid:12) e P D Ω , Σ ( x, x ; t ) − e P D Θ , Σ ( x, x ; t ) (cid:12)(cid:12)(cid:12) ≤ P x h f W t = x, e T B R < t i . (40)We can draw four lines at a distance e R := ceil( √ R ) from x such that, if f W leaves B R ( x ) , it touches at least one of the lines. The probability that thishappens for any given one of the lines is easily calculated using the reflectionprinciple, and so we have (cid:12)(cid:12)(cid:12) e P D Ω , Σ ( x, x ; t ) − e P D Θ , Σ ( x, x ; t ) (cid:12)(cid:12)(cid:12) ≤ P h f W t = (2 e R, i . (41)For t > max(1 , e R/ e ) the probability on the right hand side can be approx-imated by a Gaussian [Pan93], and this gives an estimate of the same formas obtained in Theorem 2 above for Brownian motion.In the other cases, letting N t be the number of jumps in f W up to time t ,we can estimate P x h f W t = x, e T B c R ( x ) < t i ≤ P [ N t ≥ R )] P x h f W t = x (cid:12)(cid:12)(cid:12) N t ≥ R ) i . (42) N t is Poisson distributed with mean t , so P [ N t ≥ n ] = e − t ∞ X m = n (4 t ) m m ! ≤ (4 t ) m m ! . (43)For t < this is more than enough to imply the desired bound. For t > e R we can use (4 t ) m m ! ≤ (cid:18) etm (cid:19) m = exp (cid:18) y (cid:20) − log ty (cid:21)(cid:19) ≤ e − y , (44)which together with P x h f W t = x (cid:12)(cid:12)(cid:12) N t ≥ R ) i = O (1 /R ) (45)completes the proof. In this section we review the proof of Equation (14). This is substantiallythe same proof as [Kac66], who considered the case of the trivial bundle.15laneHalf-planeInfinite wedgeFigure 3: Example of a polygonal domain Π showing the regions for which R Π ( x ) is of the specified form.We present it here in order to make it clear that it works unchanged in thenontrivial case and to bring out a number of details in a way which will makeit possible to compare with the discrete version.Let κ = κ (Π) be the such that, for all x ∈ Π , B κ ( κ ) ∩ ∂ Π is containedin the union of two adjacent line segments of ∂ Π . We will approximate P D Π ( x, x ; t ) (and, in due time, e P D L Π ( x, x ; t ) ) by the heat kernel in a referencegeometry, a set R Π ( x ) obtained by taking B κ ( x ) ∩ Π and extending infinitelyany components of ∂ Π present (see Figure 3). Note that this gives either thewhole 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 ∈ N .When R ( x ) = R , we have P D R ( x ) ( x, x ; t ) = P ( t ) = 1 / πt ; note that Z Ω P ( t ) d t = | Ω | πt = a (Ω) t − . (46)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 be takingone such line segment and displacing it perpendicularly in the direction ofthe interior of Π and removing the original line segment, so that each E ∈ E is closed on exactly one of its three sides . For E ∈ E , let ℓ ( E ) denote the This is not immediately important, but when considering the discrete version it will ℓ EE κ ℓ ( E ) = ℓH ( E ) Figure 4: Constructions related to the edge term. On the left, an example ofa polygonal region Π with one of the half-open line segments ℓ of the bondaryhighlighted. On the right, (part of) the corresponding semi-infinite rectangle E and half plane H ( E ) ; note E κ ⊂ E ⊂ H ( E ) .line segment used to construct E , w ( E ) denote the width of E (that is, thelength of ℓ ( E ) ), so that P E ∈E w ( 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 planeobtained by extending ℓ ( E ) infinitely; so the set of x with R ( x ) = H ( E ) is asubset of E κ . Some aspects of these definitions are illustrated in Figure 4.Letting x ⊥ denote the component of x perpendicular to ℓ ( E ) , we have P D H ( E ) ( x, x ; t ) = P ( t ) − πt e − x ⊥ /t =: P ( t ) + ∂P H ( E ) ( x ; t ) (47)thus Z E ∂P H ( E ) ( x ; t ) d x = − w ( E ) 18 √ π t − / (48)and thus X E ∈E Z E ∂P H ( E ) ( x, x ; t ) d x = a (Π) t − / . (49) be helpful that this has the consequence that LE , for L ∈ N , is a disjoint union of L translates of E . ( C ) ϕ ( C ) C ˆ C ˆ C E ( C ) E ( C ) v ( C ) ϕ ( C ) E ( C ) E ( C ) C Figure 5: Two examples of the notation for the definitions of the corner term; C κ is drawn in dark gray and a portion of E κ , E κ in light gray, cf. Figure 3.Note that in the example on the left E κ , E κ extend outside of C (and outsideof Π ); these extending portions are C κ , C κ .Also, (cid:12)(cid:12)(cid:12)(cid:12)Z E κ ∂P H ( E ) ( x, x ; t ) d x − Z E ∂P H ( E ) ( x, x ; t ) d x (cid:12)(cid:12)(cid:12)(cid:12) = w ( E ) 18 √ πt erfc (cid:16) κ/ √ t (cid:17) ≤ w ( E )8 πκ e − κ /t . (50)Finally, let C be the set of C ⊂ R which are open infinite wedges match-ing the corners of Π . For such a C let ϕ ( C ) be its opening angle, and let E ( C ) , E ( C ) be the elements of E associated with the line segments in-cident on the corner; thus C = H ( E ( C )) ∩ H ( E ( C )) if ϕ ( C ) ≤ π and C = H ( E ( C )) ∪ H ( E ( C )) if ϕ ( C ) > π . Also 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 edgesof C , or equivalently the set of x ∈ Π for which R Π ( x ) = C . For j = 1 , let ˆ C jκ = ( E j ) κ ( C ) \ C and let ˆ C j be the corresponding infinite wedge; both ofthese are ∅ if ϕ ( C ) ≥ π/ . 18 ( C ) C H ( E ( C )) H ( E ( C )) ϕ ( C )2 xy Figure 6: Example of the estimates on P ∠ C ( x ; t ) in the case ϕ ( C ) ≤ π/ ; here P ∠ C = P D C + P R − P D H ( E ( C )) − P D H ( E ( C )) . For the indicated x , y is both theclosest point in C △ H ( E ( C )) and the closest point in H ( E ( C )) △ R , and | x − y | ≥ sin( ϕ ( C ) / | x − v ( C ) | .The point of all this is Z Π P D R ( x ) ( x, x ; t ) d x = Z Π P ( t ) d x + X E ∈E Z E κ ∂P H ( E ) ( x ; t ) d x + X C ∈C "Z C κ P ∠ C ( x ; t ) d x − X j =1 , Z ˆ C jκ ∂P E j ( x ; t ) d x (51)(note that the integrals over ˆ C jκ cancel the part of the E κ integrals where x / ∈ Π ) where P ∠ C ( x ; t ) := P D C ( x, x ; t ) − P ( t ) , x ∈ C \ ( ˆ E ∪ ˆ E ) P D C ( x, x ; t ) − P D H ( E j ) ( x, x ; t ) , x ∈ ( C ∩ ˆ E j ) \ ˆ E k , { j, k } = { , } P D C ( x, x ; t ) − P ( t ) − ∂P H ( E ) ( x ; t ) − ∂P H ( E ) , x ∈ C ∩ ˆ E ∩ ˆ E , (52)where ˆ E j is the open quarter-plane with vertex v ( C ) obtained by extending E jκ . Using Theorem 2, this satisfies | P ∠ C ( x ; t ) | ≤ Ct − e − k | x − v ( C ) | /t for k = k (Π) = min C ∈C sin ϕ ( C ) / ; the most complicated case is illustrated in19igure 6. From Equation (47) we see that the same is true of ∂P ( x ; t ) for x ∈ ˆ C ∪ ˆ C . Consequently, we see that the integrals in a ( C ) := Z C P ∠ C ( x ; t ) d x − X j =1 , Z ˆ C j ∂P E j ( x ; t ) d x (53)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 dependsonly on ϕ ( C ) ; I will not calculate the exact form, however. We also have a ( C ) − "Z C κ P ∠ C ( x ; t ) d x − X j =1 , Z ˆ C jκ ∂P E j ( x ; t ) d x = O (cid:16) κ − e − kκ /t (cid:17) ; (54)combining this with a bunch of previous stuff and letting a (Π) = P C ∈C a ( C ) we get Z Π P D R Π ( x ) ( x, x ; t ) d x = a (Π) t − + a (Π) t − / + a (Π) + O (cid:16) e − δ /t (cid:17) . (55)Since Z Π P D Π ( x, x ; t ) d x − Z Π P D R Π ( x ) ( x, x ; t ) d x = O (cid:16) e − κ L /t (cid:17) (56)we obtain Equation (14). Let B s be a two dimensional Brownian bridge, shifted and rescaled so that B = B t = x ; we then have P D Ω , Σ ( x, x ; t ) = P ( t ) E tx (cid:20) D Ω ( B ) e i W Σ ( B ; t ) (cid:21) ; (57)where D Ω ( B ) is the indicator function of the event that B s ∈ Ω for all s ∈ [0 , t ] ; recall P ( t ) = P D R ( x, x ; t ) = 1 / πt . Similatly, letting e B s be the processobtained by conditioning the continuous time random walk f W s to return toits starting point at time t , e P D Ω , Σ ( x, x ; t ) = e P ( t ) E tx (cid:20) D Ω ( e B ) e i W Σ ( e B ; t ) (cid:21) . (58)20ombining these two expressions gives P D Ω , Σ ( x, x ; t ) − e P D Ω , Σ ( y, y ; t ) = P ( t ) (cid:26) E tx (cid:20) D Ω ( B ) e i W Σ ( B ; t ) (cid:21) − E ty (cid:20) D Ω ( e B ) e i W Σ ( e B ; t ) (cid:21)(cid:27) + " − P ( t ) e P ( t ) P D Ω , Σ ( y, y ; t ) (59)(note that the endpoints x and y are different). This is helpful because thetwo bridge processes can be coupled using a variant of the dyadic coupling,adapting a result of [LT07]: Theorem 4.
There is a constant C > such that for any y ∈ Z , x ∈ R with | x − y | ∞ ≤ / and any t > , there exists a coupling of B s and e B s suchthat P tx,y " sup s ∈ [0 ,t ] (cid:12)(cid:12)(cid:12) B s − e B s (cid:12)(cid:12)(cid:12) ≥ C log t ≤ C t − . (60) Proof.
The corresponding statement with x = y and e B s replaced by a cor-responding discrete-time random walk bridge [LT07, Corollary 3.2] followseasily from the one-dimensional construction of [KMT75]; this implies (upto a redifinition of the constant) the corresponding result for continous timeby a bound on the variance of the Poisson distribution, and then x = y bytranslation.Using this representation, E tx (cid:20) D Ω ( B ) e i W Σ ( B ; t ) (cid:21) − E ty (cid:20) D Ω ( e B ) e i W Σ ( e B ; t ) (cid:21) = E tx,y (cid:20) D Ω ( B ) e i W Σ ( B ; t ) − D Ω ( e B ) e i W Σ ( e B ; t ) (cid:21) . (61)For any δ > , restricting to the situation where the two processes remainwithin a distance δ of each other for the whole time interval [0 , t ] , the differ-ence on the right hand side can be nonzero only if both B and e B come withina distance δ of Ω c but at least one of them remains entirely in Ω . This inturn requires that in the same time interval B s leaves Ω − ( δ ) := { x ∈ Ω : d ( x, Ω c ) ≤ δ } , (62)but does not leave Ω + ( δ ) := (cid:8) x ∈ R : d ( x, Ω) < δ (cid:9) . (63)21sing this observation with δ = C log t and noting that − P ( t ) / e P ( t ) ≤ C /t , Equation (59) gives (cid:12)(cid:12)(cid:12) P D Ω ( x, x ; t ) − e P D Ω ( y, y ; t ) (cid:12)(cid:12)(cid:12) ≤ r → + πr P x (cid:2) W t ∈ B r ( x ) , T Ω − ( δ ) < t < T Ω + ( δ ) (cid:3) + C /t = 2 n P D Ω + ( δ ) ( x, x ; t ) − P D Ω − ( δ ) ( x, x ; t ) o + C /t . (64)The following estimates will be useful for estimating the first term on theright hand side of Equation (64) in different situations. Some of them aresomehwat crude versions of well known results, of which I provide basic proofsfor completeness. Lemma 5.
For all ρ, τ > and all x, y ∈ R , P y h T B c ρ ( x ) i ≤ e − τ/ ρ . (65) Proof.
For all y ∈ R , P y " sup s ∈ [0 ,ρ ] |W s − x | ≤ ρ ≤ P x [ |W ρ − x | ≤ ρ ] = 12 ρ Z ∞ ρ re − r / ρ d r = e − / , (66)and using this, P x h T B c ρ ( x ) i = P x " sup s ∈ [0 ,τ ] |W s − x | < ρ ≤ (cid:2) e − / (cid:3) floor τ/ρ ≤ e − τ/ ρ . (67)Noting that P D Ω ( x, x ; t ) ≤ P x [ T Ω c ≥ t/
2] sup y ∈ Ω P D Ω ( y, x ; t/ ≤ πt P x (cid:2) T B R ( x ) ≥ t/ (cid:3) (68)with R := dist( x, Ω c ) , this also gives Corollary 6.
For all x ∈ Ω ⊂ R , P D Ω ( x, x ; t ) ≤ e πt exp (cid:0) − t/ ( x, Ω c ) (cid:1) . (69)22ow note that for Π a polygonal domain and κ = κ (Π) introduced inSection 4, for any x ∈ ∂ Ω + ( δ ) there is a line segment of length κ − δ whichhas x as one of its endpoints and which is entirely inside of Ω + ( δ ) . We makeuse of this in the following estimates. Lemma 7.
For every τ > , Ω ⊂ R such that every point on the boundaryof Ω is the endpoint of a line segment of length √ τ contained in Ω c , P x [ T Ω c > τ ] ≤ ( e + 1) R / (log τ − R ) / τ / (70) for all x ∈ Ω with R := dist( x, Ω c ) ≤ e − / √ τ .Proof. Let y ∈ ∂ Ω be such that | x − y | = dist( x, Ω c ) =: R , and ρ := r τ log τ − R = R s τ /R log τ /R ≥ e / R (71)(the inequality follows from the assumption that R ≤ e − / √ τ ); then P x [ T Ω c > τ ] ≤ P x h T B c ρ ( y ) > τ i + P x h T Ω c > T B c ρ ( y ) i . (72)By Lemma 5 the first term is bounded by e − τ/ ρ = eR / τ − / , and notingthat ρ < √ τ the assumptions we have made imply the Beurling estimate P x h T Ω c > T B c ρ ( y ) i ≤ P x h T Λ > T B c ρ ( y ) i ≤ s Rρ , (73)where Λ ⊂ Ω c is a line segment of length ρ one of whose endpoints is y . Lemma 8.
There exists C > such that for all Ω , τ, x, R as in Lemma 7and t = 3 τ , P D Ω ( x, x ; t ) ≤ C R (log t − R ) t / . (74) Proof.
Noting that P D Ω ( x, x ; t ) = Z Ω d y Z Ω d zP D Ω ( x, y ; t/ P D Ω ( y, z ; t/ P D Ω ( z, x ; t/ ≤ (cid:20)Z Ω d yP D Ω ( x, y ; t/ (cid:21) sup z,w ∈ Ω P R ( z, w ; t/
3) = { P x [ T Ω c > t/ } πt , (75)the result follows from Lemma 7. 23 emma 9. For all α ∈ (0 , π ) , there exists C ( α ) < ∞ such that:For all polygons Π whose corner angles are in [ α, π − α ] , all δ ∈ (0 , κ (Π)) ,and all x ∈ Π \ Π − ( δ )lim r → + πr P x (cid:2) W t ∈ B r ( x ) , T Π − ( δ ) < t < T Π + ( δ ) (cid:3) ≤ C ( α ) √ tR − s δ min(2 κ (Π) − δ, R − ) e − R − / t (76) where R − = dist( x, Π − ( δ )) .Proof. lim r → + πr P x (cid:2) W t ∈ B r ( x ) , T Π − ( δ ) < t < T Π + ( δ ) (cid:3) ≤ P x (cid:2) T Π − ( δ ) < t (cid:3) lim r → + πr sup y ∈ Π − ( δ ) s ∈ [0 ,t ] P y (cid:2) W s ∈ B r ( x ) , T Π + ( δ ) > s (cid:3) (77)The first factor can be bounded as in the proof of Theorem 2, P x (cid:2) T Π − ( δ ) < t (cid:3) ≤ P x (cid:2) T B c R ( x ) < t (cid:3) ≤ √ t √ πR − . (78)As for the remaining factor on the right hand side of Equation (77), for any y ∈ Π − ( δ ) and s > we can bound lim r → + πr P y (cid:2) W s ∈ B r ( x ) , T Π + ( δ ) > s (cid:3) ≤ P y (cid:2) T B S ( w ) < T Π + ( δ ) (cid:3) sup u ∈ [0 ,s ] sup z / ∈ B R − / ( x ) P R ( z, x ; u ) (79)for any w ∈ R , S > such that S + R − ≤ | x − w | . Since y ∈ Π − ( δ ) , wecan choose a w ∈ Π + ( δ ) with | y − w | ≤ (1 + csc α ) δ and set S = min( κ (Π) − δ, R − ) and proceed as in Lemma 7 to obtain P y (cid:2) T B S ( w ) < T Π + ( δ ) (cid:3) ≤ s α ) δ min(2 κ (Π) − δ, R − ) , (80)and the last factor can be bounded sup u ∈ (0 ,s ) sup z : | z − x | >R/ P R ( z, x ; u ) ≤ sup u ≥ e − R / u πu = 4 eπR . (81)Plugging all of this into Equation (77), we obtain Equation (76).24 emma 10. For all α ∈ (0 , π ) , there exists C ( α ) < ∞ such that:For all polygons Π whose corner angles are in [ α, π − α ] , x ∈ ( L Π) − ( δ ) , δ ≤ κ (Π) L , and C ( α ) δ < R ≤ min (cid:0) e − / √ t, L (cid:1) , h P D ( L Π) + ( δ ) − P D ( L Π) − ( δ ) i ( x, x ; t ) ≤ C ( α ) (log t − R ) R + t / , R + < R √ t √ δR / , R ≤ R + < L √ t √ δL / R , R + ≥ L (82) with R + = dist( x, ∂ ( L Π) + ( δ )) .Proof. The L = 1 version of Equation (82), h P D Π + ( δ ) − P D Π − ( δ ) i ( x, x ; t ) ≤ C Π (log t − R ) R + t / , R + < R √ t √ δR / , R ≤ R + < √ t √ δR , R + ≥ (83)follows from Lemmas 8 and 9, as long as C is chosen large enough, so that R + ≤ r − whenever R + ≥ R . Noting that ( L Π) ± ( δ ) = L [Π ± ( δ/L )] , we canrescale the above inequality using P D ( L Π) ± ( δ ) ( x, x ; t ) = 1 L P D (Π) ± ( δ/L ) ( x/L, x/L ; t/L ) (84)to obtain Equation (82) for all L . Lemma 11.
For each Π with α as above, there exists C such that whenever [max( e , C ( α ) δ )] / ≤ t ≤ L / and δ ≤ κ (Π) L , Z ( L Π) − ( δ ) h P D ( L Π) + ( δ ) − P D ( L Π) − ( δ ) i ( x, x ; t ) d x ≤ C Lt / (cid:16) √ δ + log L (cid:17) . (85) Proof.
Letting R = t / , under the stated hypotheses we have C ( α ) δ For all Π there exists C < ∞ such that, whenever e / ≤ K ≤ L , Z ∞ K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z L Π P D L Π ( x, x ; t ) d x − X y ∈ L Π ∩ Z e P D L Π ( y, y ; t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d tt ≤ C (cid:18) L log LK / + L K (cid:19) . (87) Proof. Using Equation (64), we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z L Π P D L Π ( x, x ; t ) d x − X y ∈ L Π ∩ Z e P D L Π ( y, y ; t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z L Π (cid:8) P D ( L Π) + ( δ ) ( x, x ; t ) − P D ( L Π) − ( δ ) ( x, x ; t ) (cid:9) d x + C | Π | L t ≤ Z ( L Π) − ( δ ) (cid:8) P D ( L Π) + ( δ ) ( x, x ; t ) − P D ( L Π) − ( δ ) ( x, x ; t ) (cid:9) d x + 2 Z L Π \ ( L Π) − ( δ ) (cid:8) P D ( L Π) + ( δ ) ( x, x ; t ) (cid:9) d x + C L t (88)with δ = C log t . For t ≤ L / , we can bound the first integral usingLemma 11, and the second using P D ( L Π) + ( δ ) ( x, x ; t ) ≤ / πt , giving (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z L Π P D L Π ( x, x ; t ) d x − X y ∈ L Π ∩ Z e P D L Π ( y, y ; t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:18) L log Lt / + L t (cid:19) . (89)For t > L / , Lemma 7 gives a stronger bound, and Equation (87) followsimmediately. In this section I control the difference between the discrete heat kernel on Ω and in the reference geometry, and relate the latter to the terms α , α , α inEquation (5). To take into account the presence of a lattice scale, we define e P R L Π ( x ; t ) := e P D LR Π ( L − x ) ( x, x ; t ) . (90)26e now introduce counterparts of a (Π) , a (Π) , a (Π) ; the time depen-dence is not so simple as in the continuum case. For H a half-plane welet ∂ e P H ( x ; t ) := e P D H ( x, x ; t ) − e P ( t ) (91)(cf. Equation (47)) and define e P ∠ C by the analogue of Equation (52). We thenhave X Ω ∩ Z e P R L Π ( x ; t ) = (cid:12)(cid:12) Ω ∩ Z (cid:12)(cid:12) e P ( t ) + X E ∈E X x ∈ LE κ ∩ Z ∂ e P H ( E ) ( x ; t )+ X C ∈C X x ∈ LC κ ∩ Z e P ∠ C ( x ; t ) − X j =1 , X x ∈ L ˆ C jκ ∩ Z ∂ e P H ( E j ) ( x ; t ) . (92)To extract the asymptotics we are interested in, let A (Ω , t ) := (cid:12)(cid:12) Ω ∩ Z (cid:12)(cid:12) h e P ( t ) − [0 ,e − γ ) ( t ) i , (93) A ( L Π , t ) := X E ∈E ˆ A ( E, L, t ) := X E ∈E X x ∈ LE ∩ Z ∂ e P LH ( E ) ( x ; t ) , (94) A ( L Π , t ) := X C ∈C ˆ A ( C, t ) := X C ∈C X x ∈ LC ∩ Z e P ∠ C ( x ; t ) − X j =1 , X x ∈ L ˆ C j ∩ Z ∂ e P H ( E j ) ( x ; t ) . (95)Note that A is actually independent of L , A ( L Π , t ) = A (Π , t ) , while A ( L Π , t ) = LA (Π , t ) . Furthermore, A differs from the corresponding termin Equation (92) in that the sum over sites includes LE \ LE κ . We can useTheorem 3 to bound ∂ e P H ( E ) , giving X x ∈ ( LE \ LE κ ) ∩ Z (cid:12)(cid:12)(cid:12) ∂ e P H ( E ) ( x ; t ) (cid:12)(cid:12)(cid:12) ≤ C L ∞ X R =floor( Lκ ) te − R , t ≤ t − e − R , < t ≤ R/ (4 e ) t − exp ( − R / t ) , otherwise ≤ C L te − κL , t ≤ t − e − κL , < t ≤ κLt − / exp ( − κ L / t ) , t > κL. (96)Similarly, using the same observations as in Equation (54) we can estimatethe summands in Equation (95) using Theorem 3 in terms of the distance27rom the vertex v ( C ) , yielding (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ∈ LC ∩ Z e P ∠ C ( x ; t ) − X j =1 , X x ∈ L ˆ C j ∩ Z ∂ e P H ( E j ) ( x ; t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ∞ X R =floor( Lκ ) R e ρ ( R, t ) (97)for e ρ introduced in Equation (39), giving a bound of the same form; hence (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ∈ L Π ∩ Z e P D L Π ( x ; t ) − [0 ,e − γ ) ( t ) − A ( L Π , t ) − A ( L Π , t ) − A ( L Π , t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C L te − κL , t ≤ t − e − κL , < t ≤ κLt − / exp ( − κ L / t ) , t > κL ; (98)applying Theorem 3 again to bound the difference between e P R L Π and e P D L Π gives the similar bound (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ∈ L Π ∩ Z e P D L Π ( x ; t ) − [0 ,e − γ ) ( t ) − A ( L Π , t ) − A ( L Π , t ) − A ( L Π , t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C L te − κL , t ≤ t − e − κL , < t ≤ κLt − / exp ( − κ L / t ) , t > κL ; (99)and consequently (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z K ( X x ∈ L Π ∩ Z e P D L Π ( x ; t ) − [0 ,e − γ ) ( t ) − A ( L Π , t ) − A ( L Π , t ) − A ( L Π , t ) ) d tt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C K exp (cid:18) − C L K (cid:19) (100)for all √ L ≤ K ≤ L .Let us now compare these to their continuum counterparts. For A this28s quite simple: L a (Π) t − A ( L Π , t ) = L | Π | h P ( t ) + [0 ,e − γ ) ( t ) − e P i + (cid:2) L | Π | − L Π ∩ Z ) (cid:3) h e P ( t ) [0 ,e − γ ) ( t ) i = O (cid:18) L t + Lt (cid:19) , Lt → ∞ . (101)For A , we have L a (Π) t / − A ( L Π , t ) = L X E ∈E Z E ∂P H ( E ) ( x ; t ) d x − X y ∈ E ∩ Z ∂ e P H ( E ) ( y ; t ) ; (102)This difference can be bounded using a similar approach to Section 5, witha few variations and added elements.To begin with, note that ∂P H ( E ) ( x ; t ) − ∂ e P H ( E ) ( y ; t ) = P D H ( E ) ( x, x ; t ) − P ( t ) − e P H ( E ) ( y, y ; t ) + e P ( t )= P ( t ) n P ty h ∃ s ∈ (0 , t ) : e B s / ∈ H ( E ) i − P tx h ∃ s ∈ (0 , t ) : e B s / ∈ H ( E ) io + " − P ( t ) e P ( t ) ∂ e P H ( E ) ( y, y ; t ) (103)(cf. Equation (59)). The difference in probabilities on the right hand side ofEquation (103) is bounded by the probability that exactly one of the bridgeprocesses leaves the half plane H ( E ) ; as in Equation (64), we can bound thisdifference using Theorem 4, giving (cid:12)(cid:12)(cid:12) ∂P H ( E ) ( x ; t ) − ∂ e P H ( E ) ( y ; t ) (cid:12)(cid:12)(cid:12) ≤ P D [ H ( E )] + ( δ ) ( x, x ; t ) − P D [ H ( E )] − ( δ ) ( x, x ; t )+ C /t (104)for all t > , | x − y | ∞ ≤ / , with δ = C log t . This difference can bebounded as in Lemmas 7 to 9, with the simplification that the complementof [ H ( E )] + ( δ ) always contains an infinite half-line touching any point on itsboundary, giving h P D [ H ( E )] + ( δ ) − P D [ H ( E )] − ( δ ) i ( x, x ; t ) ≤ C ( (log t − R ) R + t / , R + < R √ t √ δR / , R + ≥ R (105)for all R ≤ e − / √ t . This is not enough by itself to bound the sum inEquation (102), since Equation (104) also contains a term of order /t which29oes not depend on the distance from the boundary; however, setting R = t / as in Lemma 11 gives (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ∈ E R ∩ Z ∂ e P H ( E ) ( x ; t ) − Z E R ∂P H ( E ) ( x ; t ) d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:20) log tt / + R t (cid:21) , (106)where E R is a rectangular region like E κ (cf. Figure 4), with R any numbersuch that | E R | = E R ∩ Z ) . We can use Theorems 2 and 3 to bound ∂P H ( E ) and ∂ e P H ( E ) separately, and setting R as close as possible to t / ,the contribution from E \ E R is exponentially small. In conclusion, we have (cid:12)(cid:12)(cid:12)(cid:12) A ( L Π , t ) − L a (Π) t / (cid:12)(cid:12)(cid:12)(cid:12) ≤ C L log tt / (107)for t ≥ ; note in particular that this decays faster than / √ t for t large.For the corner case, we can apply the same techniques to bound P ∠ C ( x ; t ) − e P ∠ C ( y ; t ) , giving estimated in terms of distances to the vertex v ( C ) , by thesame reasoning as in the derivation of Equation (54); so with R = t / , R = t / we have | a (Π) − A ( L Π , t ) | ≤ C " R log tt / + √ t √ δR / + R t ≤ C log tt / (108)for t large enough.We are now in a position to define α and α with the properties indicatedin the statement of Theorem 1 such that (cid:0) ( L Π) ∩ Z (cid:1) α + X x ∈ ∂ ext ( L Π ∩ Z ) α ( B ( x ) ∩ L Π)= − Z ∞ (cid:2) A ( L Π , t ) + A ( L Π , t ) + A ( L Π , t ) − a (Π) [ e − γ , ∞ ) ( t ) (cid:3) d tt ; (109)note that Equations (101), (102) and (108) imply that the above integralconverges at ∞ , and it is easy to show convergence at 0 using Theorem 3to estimate A and A and the simple bound ≤ − e P ( t ) ≤ t (cf. Equa-tion (43)) for A .Firstly, we let α := Z ∞ h [0 ,e − γ ) ( t ) − e P ( t ) i d tt = − L Π) ∩ Z ) α Z ∞ A (Ω , t ) d tt . (110)30or each E ∈ E , let e ∂ ( LE ) be the set of x ∈ Z \ LE which lie in the stripobtained by extending LE and are adjacent to at least one point of LE ∩ Z ;note that by construction exactly one of the corners of LE lies in e ∂E . Thesecorners are always in ∂ ext ( L Π ∩ Z ) , but other elements of e ∂ ( LE ) may notbe, if they are either in the vicinity of an acute corner of L Π or of a cornerwhich is within distance of a non-incident segment of the boundary of L Π (though this second case is impossible for L sufficiently large).Let ˆ α ( L Π , x ) := − X E ∈E x ∈ e ∂ ( LE ) ˆ A ( L Π , t ) n e ∂ ( LE ) ∩ Z o , (111)so that A ( L Π , t ) = − X x ∈ e ∂ ( L Π) ˆ α ( L Π , x ) (112)with e ∂ ( L Π) := S E ∈E e ∂ ( LE ) . Noting that the right hand side of Equa-tion (111) actually depends only on the slopes of the relevant edges, andcan therefore be computed knowing only the shape of L Π ∩ B ( x ) , we can let α ( B ( x ) ∩ L Π) := Z ∞ ˆ α ( L Π , t ) d tt (113)for x ∈ ∂ ext L Π which is not a corner of L Π , while for corners we can define α ( B ( x ) ∩ L Π) := Z ∞ ˆ α ( x ) + X Y L Π ( x ) ˆ α ( y ) − A ( C, t ) + a ( C ) [ e − γ , ∞ ) ( t ) (114)where Y L Π ( x ) is the set of y ∈ e ∂ ( L Π) \ ∂ ext L Π such that x is the closestcorner to y . For L sufficiently large this depends only on the slopes of thesides of L Π incident on x and can therefore is indeed a function of B ( x ) ∩ L Π .Putting this together with the other definitions, we see that Equation (109)is indeed verified. Using Equations (9) and (12), log det e ∆ L Π ,L Σ = Z ∞ ( X x ∈ L Π ∩ Z h [0 ,e − γ ) ( t ) − e P D L Π ( x, x ; t ) i) d tt . (115)31ecalling Equation (100), Z K ( X x ∈ L Π ∩ Z h e P D L Π ( x, x ; t ) − [0 ,e − γ ) ( t ) i) d tt = Z K [ A ( L Π , t ) + A ( L Π , t ) + A ( L Π , t )] d tt + O (cid:18) exp (cid:18) − C L K (cid:19)(cid:19) (116)for L/K → ∞ with K ≥ √ L , while Theorem 12 gives Z ∞ K X y ∈ L Π ∩ Z e P D L Π ( y, y ; t ) d tt = Z ∞ K Z L Π P D L Π ( x, x ; t ) d x d tt = O (cid:18) L log LK / (cid:19) (117)for L → ∞ with K ≥ √ L . Combining these expressions and recalling thedefinition of F Π in Equation (15), log det e ∆ L Π ,L Σ = − Z ∞ (cid:2) A ( L Π , t ) + A ( L Π , t ) + A ( L Π , t ) − a (Π) [ K , ∞ ) ( t ) (cid:3) d tt − Z ∞ K (cid:20) L a (Π) t − A ( L Π , t ) + L a (Π) t / − A ( L Π , t ) + a (Π) − A ( L Π , t ) (cid:21) d tt − Z ∞ K F L Π ( t ) d tt + O (cid:18) exp (cid:18) − C L K (cid:19) + L log LK / (cid:19) (118)for L/K → ∞ with K ≥ √ L , in the same regime, which, using Equa-tions (101), (107) and (108), implies log det e ∆ L Π ,L Σ = − Z ∞ (cid:2) A ( L Π , t ) + A ( L Π , t ) + A ( L Π , t ) − a (Π) [ K , ∞ ) ( t ) (cid:3) d tt − Z ∞ K F L Π ( t ) d tt + O (cid:18) exp (cid:18) − C L K (cid:19) + L log LK / (cid:19) . (119)32ewriting F L Π using Equation (22) and using Equation (14) to extend theresulting integral to zero, log det e ∆ L Π ,L Σ = − Z ∞ (cid:2) A ( L Π , t ) + A ( L Π , t ) + A ( L Π , t ) − a (Π) [ e − γ , ∞ ) ( t ) (cid:3) d tt − a (Π) Z e − γ L e − γ d tt − Z ∞ F Π ( t ) d tt + O (cid:18) exp (cid:18) − C L K (cid:19) + L log LK / (cid:19) (120)and setting K = L p C / L , using Equation (109) to rewrite the firstintegral, and setting α (Π) := − a (Π) and α (Π , Σ) := − ζ ′ ∆ Π (0) = − Z ∞ F Π ( t ) d tt (121)(recall Equation (21)) we obtain Equation (5). References [DLMF] NIST Digital Library of Mathematical Functions . http://dlmf.nist.gov/,Release 1.0.17 of 2017-12-22. F. W. J. Olver, A. B. Olde Daal-huis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark,B. R. Miller and B. V. Saunders, eds.[Aff86] Ian Affleck. “ Universal term in the free energy at a critical pointand the conformal anomaly”. 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