Winding of simple walks on the square lattice
Introduction
Counting of lattice paths has been a major topic in combinatorics (and probability and physics) for many decades. Especially the enumeration of various types of lattice walks confined to convex cones in , like the positive quadrant, has attracted much attention in recent years, due mainly to the rich algebraic structure of the generating functions involved (see e.g. [17], [7] and references therein) and the relations with other combinatorial structures (e.g. [6], [31]). The study of lattice walks in non-convex cones has received much less attention. Notable exception are walks on the slit plane [14], [18] and the three-quarter plane [16]. When describing the plane in polar coordinates, the confinement of walks to cones of different opening angles (with the tip positioned at the origin) may equally be phrased as a restriction on the angular coordinates of the sites visited by the walk. One may generalize this concept by replacing the angular coordinate by a notion of winding angle of the walk around the origin, in which case one can even make sense of cones of angles larger than 2π. It stands to reason that a fine control over the winding angle in the enumeration of lattice walks brings us a long way in the study of walks in (especially non-convex) cones.
Although the winding angle of lattice walks seems to have received little attention in the combinatorics literature, probabilistic aspects of the winding of long random walks have been studied in considerable detail [4], [5], [36], [38]. In particular, it is known that under suitable conditions on the steps of the random walk the winding angle after n steps is typically of order , and that the angle normalized by converges in distribution to a standard hyperbolic secant distribution. The methods used all rely on coupling to Brownian motion, for which the winding angle problem is easily studied with the help of its conformal properties. Although quite generally applicable in the asymptotic regime, these techniques tell us little about the underlying combinatorics.
In this paper we initiate the combinatorial study of lattice walks with control on the winding angle, by looking at various classes of simple (rectilinear or diagonal) walks on . As we will see, the combinatorial tools described in this paper are strong enough to bridge the gap between the combinatorial study of walks in cones and the asymptotic winding of random walks. Before describing the main results of the paper, we should start with some definitions.
We let be the set of simple diagonal walks w in of length avoiding the origin, i.e. w is a sequence in with for . We define the winding angle of w up to time i to be the difference in angular coordinates of and including a contribution 2π (resp. ) for each full counterclockwise (resp. clockwise) turn of w around the origin up to time i. Equivalently, is the unique sequence in such that and is the (counterclockwise) angle between the segments and for . The (full) winding angle of w is then . See Fig. 1 for an example.
Main result The Dirichlet space is the Hilbert space of complex analytic functions f on the unit disc that vanish at 0 and have finite norm with respect to the Dirichlet inner product where the measure is chosen such that . See [3] for a review of its properties. We denote by the standard orthogonal basis defined by , which is unnormalized since . With this notation for any analytic function .
For we let be the analytic function defined by the elliptic integral along the simplest path from 0 to z, where is the complete elliptic integral of the first kind with elliptic modulus k (see Appendix A for definitions and notation). The appearance of this elliptic integral in lattice walks enumeration is a natural one since is precisely the generating function for excursions of the simple diagonal walk from the origin (see (58) in Appendix A), The incomplete elliptic integral does not have a comparably simple combinatorial interpretation, but provides an important conformal mapping from a slit disk onto a rectangle in the complex plane, as detailed in Section 2.1.
For fixed k we use the conventional notation for the complementary modulus and the descending Landen transformation of k, which both take values in again (see Appendix A). Using these we introduce a family of analytic functions by setting (notice the in !) which satisfies . Even though has branch cuts at , we will see (Lemma 6, Lemma 7, Lemma 8) that has radius of convergence around 0 equal to and has finite norm with respect to the Dirichlet inner product, hence . According to Proposition 9 the norm of is given explicitly by where is the (elliptic) nome of modulus k (see (61) in Appendix A), which is analytic for k in the unit disk. Once properly normalized the family of functions provides an orthonormal basis of , i.e.
The main technical result of this paper is the following.
Theorem 1 For and , let be the set of (possibly empty) simple diagonal walks w on that start at , end on one of the axes at distance ℓ from the origin, and have full winding angle . Let be the generating function of . For fixed, there exists a compact self-adjoint operator on with eigenvectors such that Let be the subset of the aforementioned walks that have intermediate winding angles in an interval , i.e. for , and let be the corresponding generating function. If with , and or , then the generating function is related to a matrix element of a compact self-adjoint operator on with the same eigenvectors , as described in the table below. The remaining cases follow from the symmetries and , and the cases agree with the corresponding limits (using that ). The statement of (ii) remains valid for and and or as long as ℓ and p are even.
See Fig. 2 for examples of walks enumerated by Theorem 1. We emphasize that the convention for the index placement in , that is used also for other sets of walks throughout this work, is that the first index corresponds to the endpoint of the walk and the second index to the starting point. The reason for this choice is precisely the relation of the generating function to the matrix element at position of a Hilbert space operator. As we will see, composition of particular families of walks often corresponds to the composition of the respective operators (or multiplication of the respective infinite matrices). The index placement reflects the natural right-to-left ordering in the composition of these operators.
Theorem 1 is stated for fixed real values of , while from a combinatorial point of view it may be preferable to think of the generating functions as formal power series in the variable t. This raises the question to what extent the eigenvalue decomposition can be understood on the level of formal power series. To this end we prove in Proposition 10 that for any the coefficient is an analytic function in k around 0, such that we may interpret as taking values in the ring of formal power series in the variables z and t, see Section 2.2 for an explicit expansion for . With this observation Theorem 1 can be largely recast as a set of identities on formal power series. To this end we denote the multivariate generating function of by and the eigenvalues of , , by , , respectively (this includes as a special case). These eigenvalues, as given in Theorem 1(ii), are all analytic functions of t in the unit disk (see (70) and (2) for the power series representations of and ). Provided , the mth eigenvalue converges to 0 as in the formal topology of , i.e. for any the coefficient vanishes for all m large enough. Theorem 1 implies the convergent series identities in for , with α, and satisfying the respective assumptions described in Theorem 1(ii) and (iii). We leave it as an open problem whether these identities and an expression for can be derived using exclusively formal power series.
As an example of how to compute explicit generating functions with the help of Theorem 1, let us look at the set of simple diagonal walks from to that have winding angle π around the origin. According to Theorem 1(i) its generating function satisfies where we used the series expansions of and (respectively (70) and (58) in Appendix A) and that of in Section 2.2.
Application: excursions Theorem 1 can be used to count many specialized classes of walks involving winding angles. The first quite natural counting problem we address is that of the (diagonal) excursions from the origin, i.e. is the set of (non-empty) simple diagonal walks starting and ending at the origin with no intermediate returns (Fig. 3a). Actually, in this case we may equally well consider simple rectilinear walks on , thanks to the obvious linear mapping between the two types of walks (Fig. 3b). Even though walks do not completely avoid the origin, we may still naturally assign a winding angle sequence to them by imposing that the first and last step do not contribute to the winding angle, i.e. and . In Proposition 18 we prove that the generating function for excursions with winding angle is given (for fixed) by Since the summand is analytic in t around 0 and for any n, the relation implies an identity of formal power series in for any .
Similarly to Theorem 1(ii) one may further restrict the full winding angle sequence of w to lie in an open interval with , and . In this case it is more natural to also fix the starting direction, say , and we denote the corresponding generating function by . Observe in particular that . We prove in Theorem 23 that the generating function is given by the finite sum where is the “characteristic function” associated to . Since , the terms in the right-hand side of (6) corresponding to σ and are actually identical, therefore leaving only distinct terms. For non-integer values of b (see Proposition 18 for the full expression) can be expressed in closed form as where is the first Jacobi theta function (see (64) for a definition). Again (6) and (7) imply the equivalent identities on the level of formal power series in or . In Proposition 20 we prove that the power series is algebraic in t for any but that is it transcendental for . By inspecting the terms appearing in (6) we find that is transcendental if and only if or both and (see Theorem 23).
A special case of excursions for which the generating function is algebraic is and , see Fig. 4a. After removal of the first and last step of the walk and a linear transformation, these correspond precisely to walks in the quadrant starting and ending at the origin with steps in , also known as Gessel walks. Around 2000 Ira Gessel conjectured that the generating function for such walks is given (in our notation) by where is the descending Pochhammer symbol (see also Sloane's Online Encyclopedia of Integer sequences (OEIS) sequence A135404). The first computer-aided proof of this conjecture appeared in [30], and it was followed by several “human” proofs in [13], [15], [7]. Here we provide an alternative proof using Theorem 23. Indeed, we have explicitly where used that . According to our discussion above this is an algebraic power series in t, a fact about that was first observed in [12]. In Corollary 24 we deduce an explicit algebraic equation for , and check that it agrees with a known equation for .
Application: unconstrained random walks Let be a simple random walk on started at the origin. A natural question is to ask for the (approximate) distribution of the winding angle of the random walk around some point up to time j. As mentioned before, this question has been addressed successfully in the literature in the limit using coupling to Brownian motion. If , then is known [36], [4], [5], [38] to converge in distribution to the hyperbolic secant law with density (recall that ). If instead and one conditions the random walk not to hit z before time j, then converges to a “hyperbolic secant squared law” with density [36].
In Section 4 we complement these results by deriving exact statistics at finite j with the help of Theorem 1. To this end we look at the winding angles around two points in the vicinity of the starting point, namely and the origin itself. To be precise, let be the winding angle of around up to time , i.e. halfway its step from to (see Fig. 5). Similarly, let be the winding angle around the origin, ignoring the first step and with the convention that if has returned to the origin strictly before time .
If is a geometric random variable with parameter , i.e. for , then the following “discrete hyperbolic secant laws” hold (Theorem 25): where , and are explicit functions of k (see Theorem 25).
As a consequence we find probabilistic interpretations of the Jacobi elliptic functions cn and dn (see Appendix A) as characteristic functions of winding angles, Here denotes rounding to the nearest element of and we set by convention.
Since is the characteristic function of the aforementioned hyperbolic secant distribution, we may directly conclude the convergence in distribution as of the winding angle at geometric time . A more delicate singularity analysis, which is beyond the scope of this paper, would yield the same distributional limit for as , in accordance with previous work.
Application: loops The last application we discuss utilizes the fact that the eigenvalues of the operators in Theorem 1 have much simpler expressions than the components of the eigenvectors. It is therefore worthwhile to seek combinatorial interpretations of traces of (combinations of) operators, the values of which only depend on the eigenvalues. When writing out the trace in terms of the basis it is clear that such an interpretation must involve walks that start and end at arbitrary but equal distance from the origin. If the full winding angle is taken to be a multiple of 2π then such a walk forms a loop, i.e. it starts and ends at the same point.
A natural combinatorial set-up is described in Section 5. There we consider the set of rooted loops of index n, , which are simple diagonal walks avoiding the origin that start and end at an arbitrary but equal point in and have winding angle around the origin. The set naturally partitions into loops that visit only sites of even respectively odd parity ( with even respectively odd), see Fig. 6. We introduce the generating functions where we have included a factor for convenience, such that the generating functions of the set is actually . Observe that is precisely the generating function of unrooted loops of index 1, i.e. rooted loops modulo rerooting (but preserving orientation), because the equivalence class of a rooted loop w of index 1 contains precisely elements. This is not true for , since contains rooted loops w that cover themselves multiple times and thus have equivalence classes with less than elements.
Theorem 28 states that these generating functions for are given by Here is the operator with and appearing in Theorem 1(ii) whose mth eigenvalue can be obtained as the limit of the displayed expression (see the final remark of Theorem 1(ii)). Note as before that these imply the equivalent identities on the level of power series in .
A simple probabilistic consequence is the following. Let be a simple random walk on started at the origin and conditioned to return after 2ℓ steps. For each point we let the index be the signed number of times winds around z in counterclockwise direction, i.e. is the winding angle of around z. If z lies on the trajectory of , then we set . We let the clusters of index n be the set of connected components of , and for we let and respectively be the area and boundary length of component c. See Fig. 7 for an example. The expectation values for the total area and boundary length of the clusters of index n then satisfy with the asymptotics as indicated. The first result should be compared to the analogous result for Brownian motion: Garban and Ferreras proved in [28] using Yor's work [40] that the expected area of the set of points with index n with respect to a unit time Brownian bridge in is equal to . Perhaps surprisingly, we find that the expected boundary length all the components of index n (minus twice the expected number of such components) grows asymptotically at a rate that is independent of n, contrary to the total area.
Open question 1 Does , i.e. the total area of the finite clusters of index 0, have a similarly explicit expression? Based on the results of [28] we expect it to be asymptotic to as .
Finally we mention one more potential application of the enumeration of loops in Theorem 28 in the context of random walk loop soups [33], which are certain Poisson processes of loops on . A natural quantity to consider in such a system is the winding field which roughly assigns to any point the total index of all the loops in the process [24], [23]. Theorem 28 may be used to compute explicit expectation values (one-point functions of the corresponding vertex operators to be precise) in the massive version of the loop soups. We will pursue this direction elsewhere.
Discussion The connection between the enumeration of walks and the explicitly diagonalizable operators on Dirichlet space may seem a bit magical to the reader. So perhaps some comments are in order on how we arrived at this result, which originates in the combinatorial study of planar maps.
A planar map is a connected multigraph (a graph with multiple edges and loops allowed) that is properly embedded in the 2-sphere (edges are only allowed to meet at their extremities), viewed up to orientation-preserving homeomorphisms of the sphere. The connected components of the complement of a planar map are called the faces, which have a degree equal to the number of bounding edges. There exists a relatively simple multivariate generating function for bipartite planar maps, i.e. maps with all faces of even degree, that have two distinguished faces of degree p and ℓ and a fixed number of faces of each degree (see e.g. [25]). The surprising fact is that this generating function has a form that is very similar to that of the generating function of diagonal walks from to that avoid the slit until the end. A combinatorial explanation (in the particular case of critical planar maps) appears in [21] using a peeling exploration [20], [22].
If one further decorates the planar maps by a rigid loop model [9], then the combinatorial relation extends to one between walks of fixed winding angle with and planar maps with two distinguished faces and certain collections of non-intersecting loops separating the two faces. The combinatorics of the latter has been studied in considerable detail in [9], [8], [10], [21], which has inspired our treatment of the simple walks on in this paper. Further details on the connection and an extension to more general lattice walks with small steps (i.e. steps in ) will be provided in forthcoming work.
Finally, we point out that these methods can be used to determine Green's functions (and more general resolvents) for the Laplacian on regular lattices in the presence of a conical defect or branched covering, which are relevant to the study of various two-dimensional statistical systems. As an example, in recent work [32] Kenyon and Wilson computed the Green's function on the branched double cover of the square lattice, which has applications in local statistics of the uniform spanning tree on as well as dimer systems.
The author would like to thank Kilian Raschel, Alin Bostan and Gaëtan Borot for their suggestions on how to prove Corollary 24, and Thomas Prellberg for suggesting to extend Theorem 25 to absorbing boundary conditions. The author is indebted to an anonymous referee for numerous corrections and suggestions to improve the exposition. This work was supported by a public grant as part of the Investissement d'avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH, as well as ERC-Advanced grant 291092, “Exploring the Quantum Universe” (EQU). Part of this work was done while the author was at the Niels Bohr Institute, University of Copenhagen and Institut de Physique Théorique, CEA, Université Paris-Saclay.
Section snippets
Winding angle of walks starting and ending on an axis
Our strategy towards proving Theorem 1 will be to first prove part (ii) for three special cases (see Fig. 8), We define three linear operators , and on by specifying their matrix elements with respect to the standard basis in terms of the corresponding generating functions , and (with ) as The motivation to define the
Counting excursions with fixed winding angle
Recall from the introduction the set of excursions consisting of (non-empty) simple diagonal walks starting and ending at the origin with no intermediate returns. For such an excursion we have a well-defined winding angle sequence with and . Our first goal is to compute the generating function of excursions with winding angle equal to , To this end we cannot directly apply Theorem 1(i) because the excursions start and end
Winding angle distribution of an unconstrained random walk
Let be the simple random walk on started at the origin. In this section we use the results of Theorem 1 to determine statistics of the winding angle of up to time j around a lattice point or a dual lattice point . In the case that hits the lattice point z at time i, we set and for . From a physicist's point of view, we may regard the as an obstacle with absorbing boundary conditions and as one with reflecting boundary
Winding angle of loops
Another, rather interesting application of Theorem 1 is the counting of loops on . To be precise, for integer , let the set of rooted loops of index n be the set of simple diagonal walks w on that start and end at the same (arbitrary) point and have winding angle . The set naturally partitions into the even loops supported on and the odd loops on .
Theorem 28 The (“inverse-size biased”) generating
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