Winding of simple walks on the square lattice

Transcription

1 Winding of simpe was on the square attice Timothy Budd th September 07 Abstract A method is described to count simpe diagona was on Z with a fixed starting point and endpoint on one of the axes and a fixed winding ange around the origin The method invoves the decomposition of such was into smaer pieces, the generating functions of which are encoded in a commuting set of Hibert space operators The genera enumeration probem is then soved by obtaining and expicit eigenvaue decomposition of these operators invoving eiptic functions By further restricting the intermediate winding anges of the was to some open interva, the method can be used to count various casses of was restricted to cones in Z of opening anges that are integer mutipes of / We present three appications of this main resut First we find an expicit generating function for the was in such cones that start and end at the origin In the particuar case of a cone of ange 3/ these was are directy reated to Gesse s was in the quadrant, and we provide a new proof of their enumeration Next we study the distribution of the winding ange of a simpe random wa on Z around a point in the cose vicinity of its starting point, and find an intriguing probabiistic interpretation of the Jacobi eiptic functions Finay we reate the spectrum of one of the Hibert space operators to the enumeration of cosed oops in Z with fixed winding number around the origin Introduction Counting of attice paths has been a major topic in combinatorics and probabiity and physics for many decades Especiay the enumeration of various types of attice was confined to convex cones in Z, ie the positive quadrant, has attracted much attention in recent years, due mainy to the rich agebraic structure of the generating functions invoved see eg [3, ] and references therein and the reations with other combinatoria structures eg [3, 6] The study of attice was in non-convex cones has received much ess attention Notabe exception are was on the sit pane [0, ] and the three-quarter pane [] When describing the pane in poar coordinates, the confinement of was to cones of different opening anges with the tip positioned at the origin may equay be phrased as a restriction on the anguar coordinates of the sites visited by the wa One may generaize this concept by repacing the anguar coordinate by a notion of winding ange of the wa around the origin, in which case one can even mae sense of cones of anges arger than It stands to reason that a fine contro over the winding ange in the enumeration of attice was brings us a ong way in the study of was in especiay non-convex cones Athough the winding ange of attice was seems to have received itte attention in the combinatorics iterature, probabiistic aspects of the winding of ong random was have been studied in considerabe detai [8, 9, 8, 9] In particuar, it is nown that under suitabe conditions on the steps of Institut de Physique Théorique, CEA, Université Paris-Sacay timothybudd@iphtfr

2 Figure : The winding ange of a simpe diagona wa w W of ength w 9 from, 0 to, 3 The fu winding ange θ w is indicated in green, and the winding ange increment θi w θi w in bue which is negative in this exampe the random wa the winding ange after n steps is typicay of order og n, and that the ange normaized by / og n converges in distribution to a standard hyperboic secant distribution The methods used a rey on couping to Brownian motion, for which the winding ange probem is easiy studied with the hep of its conforma properties Athough quite generay appicabe in the asymptotic regime, these techniques te us itte about the underying combinatorics In this paper we initiate the combinatoria study of attice was with contro on the winding ange, by ooing at various casses of simpe rectiinear or diagona was on Z As we wi see, the combinatoria toos described in this paper are strong enough to bridge the gap between the combinatoria study of was in cones and the asymptotic winding of random was Before describing the main resuts of the paper, we shoud start with some definitions We et W be the set of simpe diagona was w in Z of ength w 0 avoiding the origin, ie w is a sequence w i w i0 in Z \ {0, 0} with w i w i {,,,,,,, } for i w We define the winding ange θi w R of w up to time i to be the difference in anguar coordinates of w i and w 0 incuding a contribution resp for each fu countercocwise resp cocwise turn of w around the origin up to time i Equivaenty, θi w w i0 is the unique sequence in R such that θw 0 0 and θi w θi w is the countercocwise ange between the segments 0, 0,w i and 0, 0,w i for i w The fu winding ange of w is then θ w θ w See Figure for an exampe w Main resut The Dirichet space D is the Hibert space of compex anaytic functions f on the unit disc D {z C : z < } that vanish at 0 and have finite norm f with respect to the Dirichet inner D product f,д D D f z д zdaz n [z n ]f z [z n ]дz, n dax + iy dxdy See [] for a review of its properties We denote by e n n the standard orthogona basis defined by e n x x n, which is unnormaized since e n D n For 0, we et v : C \ {z R : z } C be the anaytic function defined by the eiptic integra v z z dx K 0 x x aong the simpest path from 0 to z, where K 0 dx x x dy 0 y y

3 is the compete eiptic integra of the first ind with eiptic moduus For fixed we use the conventiona notation and + for the compimentary moduus and the descending Landen transformation of Using these we introduce a famiy f m m of anaytic functions by setting notice the in v z! f m z cosmv z + / cosm/ As we wi see in Proposition this famiy forms yet another orthogona basis of D, with f m D mq m q m, where q e K /K is the eiptic nome of moduus The main technica resut of this paper is the foowing 3 /, 0,3 / a W, b W 3, c W Figure : Exampes of was enumerated by Theorem /, /,, Theorem For and α Z, et Wα be the set of possiby empty simpe diagona was w on Z \ {0, 0} that start at p, 0, end on one of the axes at distance from the origin, and have fu winding ange θ w α i Let W α t w W α exists a compact sef-adjoint operator Y α t w be the generating function of W α For t 0, fixed, there W α t e, Y α e p D, Y α f m on D with eigenvectors f m m such that K m qm α / f m 3 ii Let W α,i W α be the subset of the aforementioned was that have intermediate winding anges in I R, ie θi w I for i,,, w, and et W α,i t be the corresponding generating function If I β, β + with β ± Z {± }, α [β, β + ] Z and α 0 or α β ±, then the generating function W α,i t is reated to a matrix eement of a compact sef-adjoint operator on D with the same eigenvectors f m m, as described in the tabe beow α β β + W α,i t Eigenvaues > 0 0 α > 0 < 0 α p e, A α e p A α f m K e, J α, β e p J α, β f m 0 < 0 > α e, B α, β, β + e p B α, β, β + f m m mα / mα / f m q q q mβ / q mβ / mα / q K mα / q f m q mβ / mα / q mα / m q q mβ / q mβ+/ q mβ+/ f m 3

4 The remaining cases foow from the symmetries α, β, β + α, β +, β and α, β, β + α, α β +, α β, and the cases β ± ± agree with the corresponding imits β± ± using that q 0, iii The statement of ii remains vaid for β ± Z {± } and α [β, β + ] Z as ong as and p are even For exampe, this theorem states that the set of simpe diagona was from 3, 0 to 3, 0 that have winding ange around the origin has generating function W 3,3 t e 3, Y e K e3, f m 3 D m qm D f m m D K q m 3[z 3 m q m ]f m z 0 t t t 0 + m w E q m Appication: Excursions Theorem can be used to count many speciaized casses of was invoving winding anges The first quite natura counting probem we address is that of the diagona excursions E from the origin, ie E is the set of non-empty simpe diagona was starting and ending at the origin with no intermediate returns Figure 3a Actuay, in this case we may equay we consider simpe rectiinear was on Z, thans to the obvious inear mapping between the two types of was Figure 3b Even though was w E do not competey avoid the origin, we may sti naturay assign a winding ange sequence to them by imposing that the first and ast step do not contribute to the winding ange, ie θ w θw 0 0 and θw θ w θ w In Proposition we prove that the generating function w w for excursions with winding ange α Z is given for t 0, fixed by F α t t w {θ w α } q n K q n q n α + n Simiary to Theorem ii one may further restrict the fu winding ange sequence of w to ie in an open interva I β, β + with β ± Z such that 0 I and α I Z In this case it is more natura to aso fix the starting direction, say w,, and we introduce the corresponding generating function F α,i t, such that F α,r t F α t/ We prove in Theorem that the generating function F α,i t is given by the finite sum F α,i t σα σ β+ α cos cos F t, σ, δ β + β, 8δ δ δ δ σ 0,δ Z a Diagona excursion b Rectiinear excursion Figure 3: Exampe of an excursion in diagona and rectiinear form together with its winding ange

5 where F t,b α Z F α t e ibα is the characteristic function associated to F α t For non-integer vaues of b see Proposition for the fu expression the atter can be expressed in cosed form as F t,b cos b tan b θ b, q K θ b, q, b R \ Z where θ z,q is the first Jacobi theta function see for a definition In Proposition we prove that F t,b is an agebraic power series in t for any b Q \ Z, but not for b Z By ooing at the terms appearing in we may thus deduce whether F α,i t is agebraic too As a specia case we oo at the excursions that stay in the anguar interva /, / Around 000 Ira Gesse conjectured that the generating function for such excursions is given in our notation by F 0, /, / t n0 t n+ 6 n 5/6 n / n n 5/3 n t + t + t t 8 +, where a n Γa +n/γa is the descending Pochhammer symbo see aso Soane s Onine Encycopedia of Integer sequences OEIS sequence A350 The first computer-aided proof of this conjecture appeared in [5], and it was foowed by severa human proofs in [9,, ] Here we provide an aternative proof using Theorem Indeed, we have expicity F 0, /, / t F t, 3 3 θ 3, q K t θ 3, q According to our discussion above this is an agebraic power series in t, a fact about F 0, /, / t that was first observed in [8] In Coroary 3 we deduce an expicit agebraic equation for F 0, /, / t, and chec that it agrees with a nown equation for n0 t n+ 6 n 5/6 n / n n 5/3 n Appication: unconstrained random was Let W i i 0 be a simpe random wa on Z started at the origin A natura question is to as for the approximate distribution of the winding ange θ j of the random wa around some point x,y R up to time j As mentioned before, this question has been addressed successfuy in the iterature in the imit j using couping to Brownian motion With the hep of Theorem we may compement these resuts by deriving various exact statistics at finite j A particuary appeaing statistic is obtained by formuating the probem in the foowing precise way We tae x,y /, /, and we ower the resoution at which we measure the winding ange by rounding it to the nearest integer or haf-integer mutipe of To prevent conficts from occurring in the rounding process it is natura to not oo at the winding ange θ j after the j th step, but at the winding ange θ j / haf-way the j th step or equivaenty we may set θ j / θ j + θ j Finay, we repace the fixed time j by a random geometric time ζ with parameter 0,, distributed as j j on Z 0 To be precise, we consider the we-defined random variabes {θ ζ +/} Z+ see Figure and {θ ζ /} Z, where { } A means rounding to the nearest eement of A R and by convention we set θ / 0 We prove that the characteristic functions of these variabes are exacty given by the Jacobi eiptic functions cn, and dn, of moduus with argument normaized for correct periodicity, E exp ib{θ ζ +/} Z+ cnk b,, E exp ib{θζ /} Z dnk b, Since cny, dny, sechy Ee iyη is the characteristic function of a random variabe η with the standard hyperboic secant distribution with density sechx/dx, we may directy concude the 5

6 a {θ j / } Z b {θ j / } Z+ Figure : Exampe of an unconstrained random wa The vaue of the rounded winding ange {θ j / } Z or {θ j / } Z+ around, depends on which haf pane of the universa cover of R \ {, } harbors the j th step of the wa, as iustrated by the coor shading convergence in distribution as of the winding ange at geometric time, θ ζ +/ K d η, K og as A more deicate singuarity anaysis yieds the same distributiona imit for θ j+/ / ogj as j in accordance with the probabiistic resuts of [8, 9, 8, 9], but this is beyond the scope of this paper Appication: oops The ast appication we discuss utiizes the fact that the eigenvaues of the operators in Theorem have much simper expressions than the components e p, f m of the eigenvectors It D is therefore worthwhie to see combinatoria interpretations of traces of combinations of operators, the vaues of which ony depend on the eigenvaues When writing out the trace in terms of the basis e p p it is cear that such an interpretation must invove was that start and end at arbitrary but equa distance from the origin If the fu winding ange is taen to be a mutipe of then such a wa forms a oop, ie it starts and ends at the same point A natura combinatoria set-up is described in Section 5 There we consider the set L n of rooted oops of index n, n Z, which are simpe diagona was avoiding the origin that start and end at an arbitrary but equa point in Z and have winding ange n around the origin The set L n Ln even Ln odd naturay partitions into oops that visit ony sites of even respectivey odd parity x,y Z with x + y even respectivey odd Theorem states that the corresponding generating functions for n > 0 are given by t w L n t w n w L n tr J n, n q n q n, L odd n t n q n q n, L even n t n q n q n A simpe probabiistic consequence is the foowing Let W i i0 be a simpe random wa on Z started at the origin and conditioned to return after steps For each point z R we et the index I z be the signed number of times W i i0 winds around z in countercocwise direction, ie I z is the winding ange of W i i0 around z If z ies on the trajectory of W i i0, then we set I z We et the custers C n of index n be the set of connected components of {z R : I z n}, and for c C n we et c 6

7 a L odd b L even Figure 5: Two exampes of rooted oops with different index and parity and c respectivey be the area and boundary ength of component c Then for n > 0, E c c C n E c c C n n [ ] n [ ] q n q n q n + q n n, 3 og The first resut shoud be compared to the anaogous resut for Brownian motion: Garban and Ferreras proved in [] using Yor s wor [30] that the expected area of the set of points with index n with respect to a unit time Brownian bridge in R is equa to /n Perhaps surprisingy, we find that the expected boundary ength a the components of index n minus twice the expected number of such components grows asymptoticay at a rate that is independent of n, contrary to the tota area Open question Does E [ c C 0 finite c ], ie the tota area of the finite custers of index 0, have a simiary expicit expression? Based on the resuts of [] we expect it to be asymptotic to /30 as Finay we mention one more potentia appication of the enumeration of oops in Theorem in the context of random wa oop soups [7], which are certain Poisson processes of oops on Z A natura Figure 6: A simpe wa on Z together with its custers of index ight bue and dar bue The argest custer c has area c 9 and boundary ength c 0 notice that both sides of the sit contribute to the ength 7

8 quantity to consider in such a system is the winding fied which roughy assigns to any point z R the tota index of a the oops in the process [, 0] Theorem may be used to compute expicit expectation vaues one-point functions of the corresponding vertex operators to be precise in the massive version of the oop soups We wi pursue this direction esewhere Discussion The connection between the enumeration of was and the expicity diagonaizabe operators on Dirichet space may seem a bit magica to the reader So perhaps some comments are in order on how we arrived at this resut, which originates in the combinatoria study of panar maps A panar map is a mutigraph a graph with mutipe edges and oops aowed that is propery embedded in the -sphere edges are ony aowed to meet at their extremities, viewed up to orientationpreserving homeomorphisms of the sphere The connected components of the compement of a panar map are caed the faces, which have a degree equa to the number of bounding edges There exists a reativey simpe mutivariate generating function for bipartite panar maps, ie maps with a faces of even degree, that have two distinguished faces of degree p and and a fixed number of faces of each degree see eg [] The surprising fact, for which we wi give a combinatoria expanation esewhere using a peeing exporation [6, 7], is that this generating function has a form that is very simiar to,, that of the generating function W t of diagona was from p, 0 to, 0 that avoid the sit {x, 0 : x 0} unti the end If one further decorates the panar maps by a rigid On oop mode [7], then the combinatoria reation extends to one between was of fixed winding ange W α with α Z and panar maps with two distinguished faces and a certain coections of non-intersection oops separating the two faces The combinatorics of the atter has been studied in considerabe detai in [7, 6, 5], which has inspired our treatment of the simpe was on Z in this paper Further detais on the connection and an extension to more genera attice was with sma steps ie steps in {, 0, } wi be provided in forthcoming wor Acnowedgements The author woud ie to than Kiian Rasche, Ain Bostan and Gaëtan Borot for their suggestions on how to prove Coroary 3 The author acnowedges support from the Fondation Mathématiques Jacques Hadamard This wor was initiated whie the author was at the Nies Bohr Institute, University of Copenhagen, and was supported by ERC-Advance grant 909, Exporing the Quantum Universe EQU Winding ange of was starting and ending on an axis Our strategy towards proving Theorem wi be to first prove part ii for three specia cases see Figure 7, J W,,, B W 0,,, A W, 0, We define three inear operators J, B and A on D by specifying their matrix eements with respect to the standard basis e i i in terms of the corresponding generating functions J t, B t and A t with t as e, J e p D J t, e, B e p D B t, The main reason we define the operators in this way is the foowing e, A e p p A D t Proposition The inear operators J, B, A on D are bounded and satisfy J A B 8

9 a J, J t e, J e p D b B, B t e, B e p D c A, A t p e, A e p D Figure 7: Exampes of was contributing to the three types of buiding bocs Proof It is not hard to see that for fixed t 0, A t fas off exponentiay in + p, and therefore A has finite Hibert-Schmidt norm A e p D p e p D p, p A t < In particuar, A is bounded and even compact On the other hand, the matrix eements of B with respect to the orthonorma basis e p / p p are given by B t/ p B t b p for some sequence b n n 0 that fas off exponentiay Hence, the operator norm of B is bounded by that of the symmetric Toepitz operator associated to b n n 0, which by cassica resuts is finite since b n n 0 are the Fourier coefficients of a continuous function For,m, composition of was determines a bijection between pairs of was in A,m B m,p and was in J for which the ast intersection with the horizonta axis occurs at m, 0 Hence e, J e p D J t A,m tb m,p t m m m e, A e m D em B, e p D e, A B e p D, impying that J A B, which is therefore aso bounded It is cear from the definitions that B and J are sef-adjoint, and we wi soon see in Lemma that the same is true for J The reation J A B then impies that a three operators commute Provided the operators are compact and we have aready seen this to be the case for A, they can be simutaneousy unitariy diagonaized In Section we wi diagonaize J, and then we wi chec in Section that B and A possess the same set of eigenvectors, as expected Finay, in Section 3 we wi prove Theorem by taing suitabe compositions of the operators J, B and A The operator J We wish to enumerate the was w J, p,, that start at p, 0 and end at 0, whie maintaining stricty positive first coordinate unti the end By ooing at both coordinates of the was separatey, we easiy see that these was are in bijection with pairs of simpe was of equa ength on Z, the first of which starts at p and ends at 0 whie staying positive unti the end, whie the second starts at 0 and ends at without further restrictions For fixed ength n, such was ony exist if both n p and n are non-negative even integers, in which case the Baot theorem tes us that the former was are counted 9

10 by p n n n+p/ and the atter by J t n n n+ / Therefore the generating function J t is given expicity by p n n {n p and n non-negative and even} t n 5 n n + / n + p/ It is non-trivia ony when p + is even, in which case it has radius of convergence equa to / For fixed t 0, we denote by ψ : D C the anaytic function given by ψ x x x, which maps the unit dis D bihoomorphicay onto a strict subset ψ D D It induces a inear operator R on the Dirichet space D of anaytic functions f on D that vanish at the origin by setting R f f ψ Lemma The inear operator R is bounded and J R R In particuar, J is sef-adjoint Proof Since the Dirichet norm is preserved under conforma mapping, we have R f D f ψ D f z daz f D, ψ D impying that R is bounded By the Lagrange inversion theorem one easiy finds that for n p non-negative and even one has [x n ]ψ x p Therefore n/ [z n ] z z p n/ p n [un p ] + u n e, R R e p D R e, R e p D n [x n ]ψ x [x n ]ψ x p n n n n n p n n + / n J t e, J e p D, n n + p/ n/ p n n n + p/ {n p and n non-negative and even} 6 which finishes the proof In order to diagonaize J it suffices to find an orthogona basis of D consisting of anaytic functions that are aso orthogona with respect to the Dirichet inner-product on ψ D, f,д Dψ D f z д z daz ψ D To this end we see an injective hoomorphic mapping that taes both D and ψ D to sufficienty simpe domains As we wi see the eiptic integra v z introduced in does this job First we notice that v z can be expressed in terms of the inverse function arcsn, in a suitabe neighbourhood of the origin of the Jacobi eiptic function sn, with moduus, v z z K 0 dx arcsn z, x x 7 K As depicted in Figure 8 and proved in the next emma, after the remova of two sits v maps both D and ψ D to rectanges, with the same width but different height 0

11 Figure 8: The anaytic function v maps the dis with two sits onto a rectange of width and height T K /K The shaded domains represent ψ D and its image under v Lemma v maps D \ {z R : z }, respectivey ψ D \ {z R : z }, bihoomorphicay onto the open rectange H /, / + i T /,T /, respectivey H /, / + i T /,T /, where T : K /K satisfies T T Proof It foows from eementary properties of the Jacobi eiptic function that sn, maps K, K + i K /, K / bihoomorphicay onto the disc of radius / with doube sits at {z R : z > } This shows that v indeed maps D \ {z R : z } to H The descending Landen transformation reates the Jacobi eiptic functions sn, and sn, through see eg [, 6] Inverting the reation yieds snu, + snu/ +, + sn u/ +, snu/ +, sn u, snu, ψ snu, Setting u K v x and using that K + K we observe that v ψ x v x for x D \ {z R : z } Hence, the image of ψ D \ {z R : z } agrees with that of D \ {z R : z } after substituting The presence of sits in the domain of v indicates that for any anaytic function f on D, f v can be continued to an anaytic function д on the infinite strip R + i[ T,T ], which is -periodic and satisfies д/ v дv In fact, v induces an isomorphism between D and the Hibert space H of such anaytic functions д that vanish at 0 with norm д H д v dav H A simpe Fourier series expansion now suffices to diagonaize J Proposition The operator J on D is compact and the famiy f m m defined in forms an orthogona basis of D satisfying J f m q m/ + q m/ f m, f m D m q m q m

12 Figure 9: The refection principe in the vertica axis yieds a bijection between was from p, 0 to, 0 and was p, 0 to, 0 that visit the vertica axis at east once Proof Ceary cosm + / cosm/ m forms a basis of H, which is orthogona since an expicity computation shows that cosm + / cosm/, cosn + / cosn/ H m sinhmt {mn } But then we aso have f m, f n D m sinhmt {mn } mq m q m {mn }, 8 meaning that f m m forms an orthogona basis of D with norm f m D as caimed By Lemma f m, f n Dψ D taes the same form except for the repacement, ie f m, J f n D f m, f n Dψ D m sinhmt {mn } It foows that f m is an eigenvector of J with eigenvaue f m, J f n D sinhmt f m, f n D sinhmt coshmt q m/ + q m/, where we used that T T In particuar, J is a compact operator, since /q m/ + q m/ 0 as m Notice that this verifies Theorem ii for α / and I /, / The operators B and A Reca that B t e, B e p D is the generating function for the set B of simpe diagona was from p, 0 to, 0 that maintain stricty positive first coordinate A simpe refection principe see Figure 9 teaches us that B t is given by B t B p t B p t, 9 where B m t, m Z, is the generating function of simpe diagona was from 0, 0 to m, 0 without further restrictions Lemma 3 For fixed t 0, /, B m t can be expressed as a contour integra as B m t dz i γ z m+ t z + /z, 0 where γ traces the unit circe in countercocwise direction

13 Proof The contribution of was from 0, 0 to m, 0 of ength n and m even is n n t n dz n n n + m/ i z m+ z + /z n t n n The resut then foows from summing over n 0 and reying on absoute convergence to interchange the summation and integra Based on the simiarity between the integrand of 0 and the one in the definition of v, we find the foowing usefu representation of B Lemma If f,д D are anaytic in a neighbourhood of the cosed unit dis, then K f, B д D f z f z дz дz v γ zdz, where γ traces the upper haf of the unit circe starting at and ending at Proof From the definition we see that the integrand of is continuous and bounded on the upperhaf circe, and therefore the right-hand side of converges absoutey Hence, it suffices to chec the identity for f e and д e p, p, Combining 9 and Lemma 3 we find e, Be p D B p t B p t z z z p z p dz i γ z t z + /z γ B p t B p t + B p t B +p t z z z p z p dz i γ z t z + /z, where in the ast equaity we used that both sides vanish for p + odd, whie for p + even the upper and ower haf circes contribute equay Note that z z z + z + /z z + z + /z Hence, for z on the upper-haf circe impies that v z K z z i K, z z + /z where we used that + K K Combining with we indeed reproduce the right-hand side of With the contour integra representation in hand it is now straightforward to evauate B and A subsequenty with respect to the basis f m m Proposition 3 The inear operators B and A are compact and have the same eigenvectors f m m as J satisfying B f m A f m K q m m + q m f m, m K q m/ q m/ f m 3

14 Proof The functions f m z are seen to be anaytic for z < /, and therefore we may appy Lemma to obtain K f n, B f m D f n z f n z f m z f m z v γ zdz K / 0 cosnv + it cosnv it 8K sinhnt sinhmt K sinh mt {mn } cosmv + it cosmv it dv / 0 sinnv sinmvdv Together with 8 we concude that f m is an eigenmode of B with eigenvaue f m, B f m D f m, f m D K sinh mt m sinhmt K tanhmt m K q m m + q m Since J A B is injective, we find using Proposition that A f m K m + qm q m q m/ + q m/ f m K m q m/ q m/ The eigenvaues of B and A both approach 0 as m, impying compactness f m 3 Proof of Theorem We start with part i with α Z Given a wa w Wα, et s i i N be the sequence of times at which w aternates between the axes, ie s 0 0 and for each j 0 we set s j+ inf{s > s j : θs w θs w j /} provided it exists otherwise s j s N is the ast entry in the sequence Let α j j0 N and j j0 N be the sequences of winding anges and distances to the origin defined by α j θs w j respectivey j w sj for 0 j N It is now easy to see that for 0 j < N the part of the wa between time s j and s j+ is up to a unique rotation around the origin and/or refection in the horizonta axis of the form of a wa w j J j+, j Simiary, the ast part of the wa between time s N and w is up to rotation of the form of a wa w N B,N See Figure 0 for an exampe In fact, this construction is seen to yied a bijection between W α and the set of tupes N, j j0 N, α j j0 N, w j j0 N where N 0, 0 p, j, α j j0 N is a simpe wa on Z from α 0 0 to α N α, w j J j+, j for 0 j < N and w N B,N If we denote by a α N N N α {N α even and non-negative} the number of simpe was on Z from 0 to α of ength N 0, then we may identify the generating

15 a b c / Figure 0: a A wa w W 5,3 ; b its decomposition into w 0 J,3, w,w J,, w 3 B 5, ; c its winding ange sequence θi w function of W α as W α t N 0 N 0 N 0 a α N a α N a α N,, N B,N t N i0 J i+, i t N e, B e N D ei+, J e i D,, N i0 i+ e, B J N e p D Since the eigenvaues of J are a stricty smaer than / and a α N N, N 0 a α N B J N a compact sef-adjoint operator Y α satisfying converges to With a itte hep of 6, we find the forma generating function a α x N 0 W α t e, Y α e p D 3 a α N x N x Then one may deduce after some simpification that Y α f m B a α J f m K K x x q m m + q m a α α m/ q f m m, q m/ α / + q m/ in agreement with part i of Theorem We coud easiy extend this resut to the case I β, β + with β ± Z {± } such that 0 I and α I Z, by repacing aα x by the generating function of simpe was on Z confined to an interva Instead, we choose to discuss a refection principe at the eve of the simpe diagona was, which aows us to directy generaize to the case of β ± Z {± } of part iii f m 5

16 a b / Figure : Exampe with α 0, I /, / a A wa w W,6 is depicted in bac, and its refection w W 0,6 in gray b The corresponding winding ange sequences θw i and θi w Lemma 5 Suppose and p are both odd and β ± Z, or and p are both even and β ± Z, such that 0 I If in addition α Z I, then the generating function for Wα,I is given by W α,i t n α +nδ W t W β + α +nδ t, δ β + β Proof Consider any wa w n W β + α +nδ and et s inf{j 0 : θ w j I } be the first time w eaves I, which is we-defined since θ w I It is not hard to see that under the stated conditions on, β ± the winding ange sequence θi w w i0 cannot cross β ± without visiting β ±, and therefore θs w β ± and w s ies on an axis or a diagona of Z Then we et w be the wa obtained from w by refecting the portion of w after time s in this axis or diagona see Figure Then θ w β + θ w or θ w β θ w Hence w α +nδ n W It is not hard to see that this mapping w w is injective the inverse w w is given by the exact same refection operation Moreover, any wa w n W obtained in such way provided θi w w the atter condition are the ones in W α,i α +nδ i0 visits β ± at east once Ceary, the ony was w not satisfying The caimed resut for the generating function W α,i t t < readiy foows absoute convergence is granted because α Z W α Inspired by this resut et us introduce for β ± Z such that 0 I and α Z I, the operator on D defined by B α, β, β + B α, β, β + n Y α +nδ Y β + α +nδ, δ β+ β By Theorem i it is we-defined, compact and sef-adjoint and has eigenvaues K m n Lemma 5 then tes us that q α +nδ m/ q β + α +nδ m/ K W α,i t e, B α, β, β + e p D q mβ / mα / m q mα / q q mβ +/ q mβ / q mβ +/ is 5 hods under the conditions stated in the emma, which exacty verifies part ii and iii for 0, α I and β ± finite 6 6

17 Simiary when β or β + one may introduce the operators B α,, β + Y α Y β + α and B α, β, Y α Y β α It is straightforward to chec that then 6 sti hods, and that the eigenvaues are given by 5 in the appropriate imit β or β + Next, et us consider the case 0 < α Z and I 0, α The case α / with the corresponding operator A A / α, 0,α has aready been setted above, so et us assume α Any such wa w W is naturay encoded in a tripe w,w,w 3 of was with w A,p, w α, /,α / W and w 3 A, for some, Hence α, 0,α W t, p e, A e D α, /,α / e, B e D α, /,α / e, A B A e p D α, /,α / One may easiy verify the caimed eigenvaues of A α A B Finay, a simiar argument shows that for 0 < α Z, 0 > β Z, W α, β,α t e, A e D e, B α /, β,α e p D, e, A e p p D A and its compactness e, A B α /, β,α e p D, and once again one may directy verify the eigenvaues of A B α /, β,α This finishes the proof of Theorem 3 Excursions Reca from the introduction the set of excursions E consisting of non-empty simpe diagona was starting and ending at the origin with no intermediate returns For such an excursion w E we have a we-defined winding ange sequence θi w w i0 with θw θw 0 0 and θw θ w θ w Our first goa w w is to compute the generating function of excursions with winding ange equa to α Z, F α t t w {θw α } 7 w E To this end we cannot directy appy Theorem i because the excursions start and end at the origin Nevertheess, a combinatoria tric aows us to reate F α t to the generating functions W α with α > α Lemma 6 For α Z, F α t may be expressed as the absoutey convergent sum F α t m, +p+m+ mw α +m / Proof For the absoute convergence it suffices to note that / W t / t and for n > 0 W n / t W / t n t t n Next we use that for α Z >0, the sets p Z >0 W α and p Z >0 W α are neary in bijection Indeed, a wa in the former is mapped to a unique wa in the atter by moving its starting point by ±, 0 depending on the direction of the first step see Figure, whie eeping the other sites fixed 7

18 Figure : By changing the starting point the dar bue wa in W, is mapped to a wa in W, The dar green wa w W, satisfies w w w, such that moving its starting point and endpoint to the origin gives an excursion It is not hard to see that any wa p Z >0 W α is obtained in such way except for those was in W α, that have w, ± The generating function of such was is therefore given by w W α, An anaogous argument for the endpoint then yieds S α t w W α, t w { w } p+ W α p t w { w w w } t p+ W α t 8 Let X be the set of four possibe tupes w,w w, θ w w θw, then we may write S α t t w {w x,w w y } t w {w x,w w y } {θ w α }, x,y,α X w W α, p, x,y,α X w E where we used the obvious mapping to excursions by merey moving the starting point and endpoint to the origin see Figure By the rotationa symmetry of the set of excursions we may repace the {w x,w w y } by / in the ast sum By observing that α taes exacty the four vaues α /, α, α, α + /, one finds that S α t x,y,α X w E t w {θ w α } F α / t + F α t + F α + / t From here it is an easy chec by substitution that for α Z >0, m+ m S α +m / t F α t F α t, m which together with 8 yieds the caimed identity We are now in the position to appy Theorem i and expicity evauate F α t, as we as its characteristic function F t,b t w e ibθ w F α t e ibα w E α Z 8

19 Proposition The excursion generating functions are given by F α t q n K q n q n α +, 9 n cos b b θ b, q K θ b, q for b R \ Z, F t,b K for b Z, 0 E for b Z +, + E K for b Z +, where K and E are the compete eiptic integras of the first and second ind, and θ z,q is the Jacobi theta function θ z,q n q n+/ sinn + z n0 Proof Combining Lemma 6 with Theorem i we find F α t m, K K +p+m+ m α +m / e, Y m,n n n m+m n qn q n α + + q n e p D α +m f n p f n, e p p f n p f n, e p p D D, where we used that f n, e p D 0 for n odd Since f m z has radius of convergence arger than one, we have p f n, e p p p [z p ]f D n z i f n i f n i p p From the definition of v one may read off that v i /K + /K Together with we then find p e p, f n n K n sinhnt n K nq n q n p Combining with and 8 we arrive at F α t K n q n q n q + q n n α + q n q n K q n n q n q n α + Using the identity α Z x α e ibα x x x + x cosb/ for 0 < x < 9

20 we obtain with some agebraic manipuation F t,b K K K q n n q n cosb/ q n q n + q n q n + q n q n + q n cosb/ [ cosb/ q n + q n cosb/ + cosb/ n n q n + q n 3 ], 5 where the ast equaity ony hods for b R \ Z + The first term in the sum can be handed for b R \ Z by recognizing that q n + q n cosb/ n n m sinb/ q mn sinbm/ sinb/ q m q m m [ θ b/, q θ b/, q cotb/ sinbm/ sinb/ ], where in the ast equaity we used [, 69] The second term foows from [, 73], n q n + q n + K Hence, for b R \ Z we have F t,b cos b tan b θ b, q K θ b, q It remains to chec the vaue of F t,b at b 0,,, since F t,b + F t,b and F t, b F t,b Starting from or 5 and using [5, 3 and 6] we obtain F t, 0 F t, F t, K K K K K n q n + q n K, q n n + q n q n n + q n [ K K n q n/ + q n/ q n n + q n q n n + + q n n q n / + q / n n [ EK K K ] ] EK, q n + q n The compete eiptic integras of the first and second ind, K and E, are we nown not to be represented by agebraic power series in Hence, the same is true for F t,b at integer vaues of b The situation is surprisingy different at other rationa vaues of b: Coroary t F t,b is agebraic for a b Q \ Z 0

21 Proof Using the Landen transformation θ u,q/θ u,q+θ u,q/θ u,q θ u, q/θ u, q, which foows eg from the series representations [, 69 & 69], we may rewrite F t,b as F t,b cos b tan b θ b,q K θ b,q + θ b,q θ b,q, which in turn can be expressed using Jacobi s zeta function Z u, see [, 63 & 63] as [ ] b F t,b cos cn u, dn u, tan Z u, +, u K b/ 6 b sn u, The trigonometric functions as we as the eiptic functions cn,, dn, and sn, are we-nown to be agebraic at rationa mutipes of their period, due to the existence of agebraic mutipe-ange formuas The same is true for Z u, since it satisfies an addition formua [, 735], Z u + v, Z u, + Z v, snu, snv, snu + v,, 7 and Z K n, 0 for a n Z Hence, for any rationa vaue of b one can express Z K b/, as a poynomia in sn, evauated at rationa mutipes of its period, which are agebraic in Coroary Consider a simpe random wa on Z started at the origin The probabiity that its winding ange around the origin upon its first return equas m/ for m Z is m + m + m + 3 ψ + ψ ψ, where ψ x Γ x/γx is the digamma function Proof Let m Z be fixed First we rewrite the sum in 9 as q n n q n n m + q n p0 p0 As q the atter is asymptoticay equa to n m +p+ q q n m +p+ q + m +p+ q q m +p+3 [ ] m + p + m + p O m + p + 3 q p0 m + m + m + 3 ψ + ψ ψ + O, q where we used the series representation ψ x + ψ p p x+p of the digamma function Since K K /ogq / q + O, we find from 9 that as t /, F m / t m + m + m + 3 ψ + ψ ψ From the definition 7 it is easiy seen that this is precisey the desired probabiity Before me move on et us have a oo at the asymptotics of F t,b Thans to Kiian Rasche for pointing out this reation!

22 Lemma 7 For fixed b [0, ] the coefficients of t F t, b satisfy the estimate Γ+b sin b + b for b 0, b+ [t ]F t,b og for b 0 for b 3 The anaogous resuts for other b R foow from F t,b F t,b + F t, b Proof Genera properties of the various functions invoved indicate that for fixed b R, F /,b is anaytic in a domain that incudes {z C : z < + ϵ} \ {z R : z } for some ϵ > 0 Hence, we shoud focus on the behaviour of F /,b as For b fixed we may use Jacobi s identity see eg Ahiezer, 6 b θ, q i e 3T b θ i b,q T 8T Taing ogarithmic derivatives in b on both sides, we find θ b, q K θ b, i q K K K i θ θ i b 8T,q + b 8T θ i b 8T,q 6T i b 8T,q + b θ i b 8T,q From the definition it is cear that θ z,q /θ z,q cotz + O sinzq and K / + O, therefore for 0 < b < we find θ b, q b θ b, q b + coth 8T b + q b/ + q b/ q b/ q b/ + O + O b sinh q T q b q b q b + qb + O q b + q b Hence, for b 0,, as ± we find F t,b + b tan b cos tan b b cos q b b + O q b + q b Since q /6 + O, the dominant singuarity is proportiona to b and then transfer theorems tes us that tan b [t ]F t,b b cos b Γ + b + b sin b Γ b b+ b+, where we used the refection formua Γ zγz / sinz One may chec that this formua is vaid for b as we, since F t,b E/ has dominant singuarity og /, and therefore [t ]F t, / For b, however F t, has singuarity og /8 and [t ]F t, / 3, which differs by a factor of two from the genera formua Finay, for b 0 we may use that [ ]F, 0 is the probabiity that a simpe random wa first returns to the origin at time, which is we nown to be asymptoticay proportiona to / og The constant term is exacty the characteristic function of the probabiity distribution in Coroary

23 3 Excursions restricted to an anguar interva We turn to the probem of enumerating excursions w with winding ange sequence θi w w i0 restricted to ie fuy within an interva I R For convenience we et E {w E : w, } be the set of excursions that eave the origin in a fixed direction Then we et F α,i t be the generating function F α,i t t w {θ w α and θi w I for 0 i w } w E Notice that with this definition F α,r t F α t/ Theorem For I β, β +, β ± Z such that 0 I, and α I Z the generating function F α,i is given by the finite sum F α,i t σα σ β+ α cos cos F t, σ, δ β + β 8 8δ δ δ δ σ 0,δ Z It is agebraic if β + β Z +, or if β ± Z and either β + β Z + or α Z + or β + α Z Moreover, its coefficients satisfy the asymptotic estimate α [t β+ α ]F n,m,p t cos cos sin Γ + δ + /δ δ δ δ δ + /δ Proof By a refection principe that is competey anaogous to that used in Lemma 5 we observe that F α,i t is given by the sum F α,i t n F α +nδ t F β + α +nδ t, δ β + β, where the factor / is due to the fourfod difference between E and E Then the discrete Fourier transform e iσ α /δ e iσ β + α /δ e iσ α /δ {α α δ δ Z} {β+ α α δ Z} eads to σ 0,δ Z F α,i t 8δ σ 0,δ Z e iσ α /δ e iσ β + α /δ α Z F α te iσ α /δ Using that F t, σ/δ F t, δ σ /δ, the atter is seen to agree exacty with 8 Since F α,i t is expressed in terms of trigonometric functions at rationa anges and F t,b at rationa vaues of b, it foows from Coroary that F α,i t is agebraic if the sum does not invove F t,b at integer vaues ofb This is certainy the case when δ Z+, since σ/δ Z for σ 0, δ Z Suppose now β ± Z Then cos σα/δ cos σ β + α/δ 0 for σ δ/, meaning that F t, does not contribute to F α,i t Hence, the ony remaining obstruction for agebraicity is the case that δ σ and cosα cosβ + α 0, which is evaded by having either δ Z + or α Z + or β + α Z According to Lemma 7 the asymptotics of F α,i t is determined by the terms σ / and σ δ / If δ > then these are distinct and equa, whie in the case δ we noticed that the asymptotics for F t, incudes an additiona factor of two compared to the formua for b 0, Hence, in genera as, [t ]F α,i t δ cos in accordance with the caimed resut α β+ α cos δ δ 3 sin δ Γ + δ + /δ + /δ,

24 a b Figure 3: Excursions a of ength n + staying in the anguar interva /, / are reated to was b in the quadrant starting and ending at the origin with n steps in {0,, 0,,,,, } As a specia case we oo at excursions that stay in the anguar interva /, /, see Figure 3 Coroary 3 Gesse s attice path conjecture The generating function of excursions that stay in the anguar interva /, / with winding ange α 0 is F 0, /, / t F t, 3 θ 3, q 3 K t θ 3, q t n+ 6 n 5/6 n / n 9 n0 n 5/3 n Proof The first two equaities foow from Theorem and Proposition respectivey It remains to show that our generating function reproduces the nown formua n0 t n+ 6 n 5/6 n / n [F n 5/3 n, 6 ; ] 3 ; t, which we wi do by showing they both sove the same agebraic equation and checing that the first few terms in the expansion agree Denoting y F t, /3 + and using 6 we get with t y cn u, dn u, K 3 Z u, +, u sn u, 3 Appying the addition formua 7 to Z u + u + u, twice one finds 0 Z 3u, 3Z u, snu, sn u, + snu, sn3u, snu, 3Z u, snu, sn u,, where we used that Z K, snk, 0 Using the doube argument formua [, 68] snu, snu, cnu, dnu, sn u, as we as dn u, cn u, sn u, see [, 69], one may express y in terms of x dnu, as y x x + 3 x + x 3 x 30 Using the various addition theorems appied to snu +u +u, 0 and rewriting in terms of x dnu, one finds after a sighty tedious cacuation that x soves x + x3 + x

25 Eiminating from 30, y is then seen to be given by y 3 x 3 + x Finay one may chec that these sove 7y 8 608t + 03t + 8 y t t + t 8 y 65536t 8 688t t 8t 0, which is equivaent to the poynomia in [8, Coroary ] after substituting t t, T y /t Jacobi eiptic functions are characteristic functions Theorem 3 For a simpe random wa on Z started at the origin et θ j+/ be its winding ange around, up to haf-way between its jth and j + th site and et θ / 0 by convention Let aso ζ be a geometric random variabe with parameter, ie having distribution j j on Z 0 Then we have the hyperboic secant aws P [ n θ ζ / < n + ] P [ n θ ζ +/ < n + ] K sech K sech n K, K n + K K If we denote by { } A : R A rounding to the cosest eement of A R conficts do not arise here then the corresponding characteristic functions are given by ] E exp [ib{θ ζ /} Z dnk b,, 3 ] E exp [ib{θ ζ +/} Z+ cnk b,, 3 where cn, and dn, are Jacobi eiptic functions with moduus In order to approach this probem we require a new buiding boc, in the sense of Section, that invoves was that start on an axis but end at a genera point In particuar we wi consider the set C n of possiby empty was w starting at n, 0 and staying stricty inside the positive quadrant, ie w i Z >0 for i w, with generating function C n t Lemma 8 For m positive and odd we have C n t e n, f m D 8 K n mq m q m q m/ + q m/ Proof Let us consider the set of a possiby empty diagona was starting at p, 0 for p positive and odd, which has generating function / Some of these was visit the vertica axis and this necessariy happens away from the origin because p is odd By decomposing such was at their first visit see Figure a we find that they have generating function J t e, J e p D, 5

26 a b Figure : a An unconstrained wa starting at, 0 that visits the vertica axis decomposes into a wa in J 3, in bac and another unconstrained wa in bue b A wa starting at, 0 and avoiding the vertica axis decomposes into a wa in B 5, in green and a wa in C 5 in bac where the factor taes into account that the first visit may occur at the positive or negative side of the vertica axis Hence, the remaining was, those starting at p, 0 and avoiding the vertica axis, have generating function e, J e p D e, I J e p By decomposing the atter was at their ast intersection with the horizonta axis see Figure b, we may aso express their generating function as e, I J e p D C n e n, B e p D Using that f m, e p 0 ony when m +p is even, this impies with the hep of Propositions 3 and that D n C n e n, f m D m + q m K q m K n C n e n, B f m D n m + q m q m m + q m K q m e, I J f m D q m/ + q m/ Since f m has a radius of convergence arger than one, we may evauate e, f m D [z ]f m z f m coshmt q m/ D e, f m D + q m/ Combining the ast two dispays yieds the caimed resut Proof of Theorem 3 Let α Z We observe that P [ α θ ζ +/ < α + ] G α /, 6

27 a b Figure 5: a An unconstrained wa w with θw w +θw 3/, b Its decomposition w into w 3 / W 3, bue, w C 3 bac and s, green where G α t is the generating function for the set of non-empty simpe diagona was w that start at, 0 and have winding ange θw + θ w w w α, α + see Figure 5a It is not hard to see that such a wa can be uniquey encoded in a tripe w,w, s, where w α / α + / W n, W n, for some odd n, w C n and s {,,,,,,, } a singe step see Figure 5b As a consequence we obtain the reation G α t t n With the hep of Theorem i this evauates to For m odd we have G α t n m,n m,n C n t α / α + / W n, t + W n, t C n t α / e n, Y + Y C n t e n, f m D C n t e n, f m D α + / e D α / α + / e, Y + Y f m D f m D 8K m α / q q m/ + q m/ m q m q m e, f m D e, f m D f m 0 m+/ m v 0 m+/ m K m+/ m, K whie e, f m D 0 for m even Together with Lemma 8 and some manipuation this eads to G α t K p p0 q p+/ p+ α / q + q p+/ In particuar, the probabiity that θ ζ +/ n, n + for n Z is then P [ n θ ζ +/ < n + ] G n + / / + G n +3 / / p p+ n +/ q K K p0 q n+/ + q n / K sech n + K K 7

28 Simiary, for n Z \ {0} we have P [ n θ ζ +/ < n + ] G n / / + G n + / / p p+ n q K K q n + q n K sech n K K p0 According to [5, 9], n sechnk /K K / and therefore But then for genera n Z, P [ θ ζ +/ < ] + K P [ n θ ζ / < n + ] {n0} + P [ n θ ζ +/ < n + ] K sech n K K With the hep of [, 63 & 633] the characteristic functions may finay be expressed as E exp ib{θ ζ /} Z E exp ib{θ ζ +/} Z+ K K n n e ibn q n + q n eib n+/ q n+/ dnk b,, + q n / cnk b, 5 Winding ange of oops Another, rather interesting appication of Theorem is the counting of oops on Z To be precise, for integer n 0, et the set L n of rooted oops of index n be the set of simpe diagona was w on Z \ {0, 0} that start and end at the same arbitrary point and have winding ange θ w n The set L n Ln even Ln odd naturay partitions into the even oops Ln even supported on {x,y Z : x + y even, x,y 0, 0} and the odd oops Ln odd on {x,y Z : x + y odd} Theorem The inverse-size biased generating functions for L even n L even n t L odd n t w L even n w L odd n t w w t w w n tr D P even n, J n n tr D P odd n, J n and L odd n q n q n q n q n are given by where P even respectivey P odd is the projection operator onto the even respectivey odd functions in D Proof Without oss of generaity we wi tae n > 0, since the case of negative n then foows from symmetry Let us consider the subset,, ˆL n {w L n : w 0 {x, 0 : x > 0} and θ w i < θ w for 0 i < w } of rooted oops of index n > 0 that start on the positive x-axis and that attain the winding ange n ony at the very end Ceary ˆL n n,, n p W in the notation of Theorem Simiary p,p 8

29 ˆL even/odd n ˆL n L even/odd p even/odd W are therefore given by t w p even/odd w ˆL n p even/odd n,, n p,p The generating functions of n, ep, J e p tr D D P even/odd J n, m even/odd ˆL even n and q m+n, ˆL odd n where we used that according to Theorem ii J n, has eigenvaues q nm m and that the even and odd subspaces of D are spanned by the even respectivey odd eements of the basis f m m Hence, w ˆL even n t w qn q n and w ˆL odd n t w qn q n 33 Now suppose we tae a genera oop w L n We denote by w j L n, j w, the cycic permutation of w given by the wa w j w j,w j+,,w w,w,,w j We caim that among these w cycic permutations are exacty n eements of ˆL n To see this, et i m be the sequence of increasing times in {,,, w } at which w intersects the positive x-axis Then w i, m, are potentia candidates for was in ˆL n, since they start on the positive x-axis For each such wa w w i we may consider the winding ange sequence θi w w i0 as we as the subsequence α j m j0 of θw i w i0 containing just those anges in Z Then α j m j0 describes a wa on Z from 0 to n with steps in {, 0, }, and w i ˆL n precisey when this wa stays stricty beow n unti the very end Since the was α j m j0, m, correspond precisey to an equivaence cass under cycic permutation of the increments, a we-nown cyce emma or baot theorem tes us that the atter condition, hence w i ˆL n, is satisfied for exacty n vaues of The caim impies that t w w n w L n t w w w L n w {w j ˆL n } n j w ˆL n t w w w {w j L n } t w n j w ˆL n This identity restricted to the even and odd subspaces together with 33 then gives the desired expressions For the rest of this section we switch to simpe rectiinear rooted oops on Z For such a oop w we et the index I w : R Z be defined by setting I w z 0 when z ies on the trajectory of w and otherwise I w z is the winding ange of w around the point z By a suitabe affine transformation we may now equay thin of Ln odd, n 0, as the set of simpe rooted oops on Z with index I w z with respect to some fixed off-attice point, say, /, / Simiary, Ln even, n 0, is in -to- correspondence with such oops that have index n with respect to a fixed attice point, say, the origin The foowing probabiistic resut taes advantage of this point of view Coroary For, et W W i i0 be a simpe random wa on Z conditioned to return to the origin after steps For n 0, et C n be the set of connected components of I W n Then E c c C n E c c C n n [ ] n [ ] where c is the area of c C n and c the boundary ength of c 9 q n q n q n + q n n, 3 og,