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Winding of Simple Walks on the Square Lattice

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Winding of Simple Walks on the Square Lattice Winding of simple walks on the square lattice Timothy Budd∗ 5th February 2020 Abstract A method is described to count simple diagonal walks on Z2 with a xed starting point and endpoint on one of the axes and a xed winding angle around the origin. The method involves the decomposition of such walks into smaller pieces, the generating functions of which are encoded in a commuting set of Hilbert space operators. The general enumeration problem is then solved by obtaining an explicit eigenvalue decomposition of these operators involving elliptic functions. By further restricting the intermediate winding angles of the walks to some open interval, the method can be used to count various classes of walks restricted to cones in Z2 of opening angles that are integer multiples of π/4. We present three applications of this main result. First we nd an explicit generating function for the walks in such cones that start and end at the origin. In the particular case of a cone of angle 3π/4 these walks are directly related to Gessel’s walks in the quadrant, and we provide a new proof of their enumeration. Next we study the distribution of the winding angle of a simple random walk on Z2 around a point in the close vicinity of its starting point, for which we identify discrete analogues of the known hyperbolic secant laws and a probabilistic interpretation of the Jacobi elliptic functions. Finally we relate the spectrum of one of the Hilbert space operators to the enumeration of closed loops in Z2 with xed winding number around the origin. Keywords: Lattice walks; Random walks; Winding angle; Generating functions; Elliptic functions 1 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 conned to convex cones in Z2, 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 [15,5] and references therein) and the relations with other combinatorial structures (e.g. [4, 31]). The study of lattice walks in non-convex arXiv:1709.04042v5 [math.CO] 4 Feb 2020 cones has received much less attention. Notable exception are walks on the slit plane [12, 16] and the three-quarter plane [14]. When describing the plane in polar coordinates, the connement of walks to cones of dierent 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 ne 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 com- binatorics literature, probabilistic aspects of the winding of long random walks have been studied in ∗Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen. [email protected] 1 Figure 1: The winding angle of a simple diagonal walk w 2 W of length jwj = 19 from ¹2; 0º to (−1; 3º. The (full) winding angle θw is indicated in green, and the winding angle increment w w θi − θi−1 in blue (which is negative in this example). considerable detail [21, 22, 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 logn, and that the angle normalized by 2/logn 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 Z2. 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 denitions. We let W be the set of simple diagonal walks w in Z2 of length jwj ≥ 0 avoiding the origin, i.e. w jw j 2 is a sequence ¹wi ºi=0 in Z n f¹0; 0ºg with wi − wi−1 2 f¹1; 1º; ¹1; −1º; (−1; −1º; (−1; 1ºg for 1 ≤ i ≤ jwj. w We dene the winding angle θi 2 R of w up to time i to be the dierence in angular coordinates of wi and w0 including a contribution 2π (resp. −2π) for each full counterclockwise (resp. clockwise) turn of w jw j w w around the origin up to time i. Equivalently, ¹θi ºi=0 is the unique sequence in R such that θ0 = 0 w w and θi − θi−1 is the (counterclockwise) angle between the segments ¹¹0; 0º;wi−1º and ¹¹0; 0º;wi º for ≤ ≤ j j w w 1 i w . The (full) winding angle of w is then θ B θ jw j. See Figure1 for an example. Main result The Dirichlet space D is the Hilbert space of complex analytic functions f on the unit 2 disc D = fz 2 C : jzj < 1g that vanish at 0 and have nite norm k f kD with respect to the Dirichlet inner product ¹ 1 0 0 Õ n n hf ;giD = f ¹zºg ¹zºdA¹zº = n »z ¼f ¹zº »z ¼g¹zº; D n=1 ¹ º 1 ¯ ¹ º where the measure dA z1 + iz2 B π dz1dz2 is chosen such that D dA z = 1. See [3] for a review 1 n of its properties. We denote by ¹enºn=1 the standard orthogonal basis dened by en¹zº B z , which 2 n is unnormalized since ken kD = n. With this notation hen; f iD = n»z ¼f ¹zº for any analytic function f 2 D. 2 For k 2 ¹0; 1º we let vk : C n fz 2 R : z ≥ kg ! C be the analytic function dened by the elliptic integral 1 ¹ z dx ¹ º vk z B p (1) 4K¹kº 0 ¹k − x2º¹1 − kx2º along the simplest path from 0 to z, where K¹kº is the complete elliptic integral of the rst kind with elliptic modulus k (see AppendixA for denitions and notation). The appearance of this elliptic integral 2 in lattice walks enumeration is a natural one since π K¹4tº is precisely the generating function for 2 excursions of the simple diagonal walk from the origin (see (58) in AppendixA), 1 2 Õ 2n2 K¹4tº = t2n : (2) π n n=0 The incomplete elliptic integral vk 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 xed k we use the conventional notation p 1 − k 0 k 0 = 1 − k2 and k = 1 1 + k 0 0 for the complementary modulus k and the descending Landen transformation k1 of k, which both take 1 values in ¹0; 1º again (see AppendixA). Using these we introduce a family ¹fmºm=1 of analytic functions by setting (notice the k1 in vk1 ¹zº!) fm¹zº B cos¹2πm¹vk1 ¹zº + 1/4ºº − cos¹πm/2º; (3) p which satises f ¹0º = 0. Even though v ¹zº has branch cuts at z = ± k , we will see (Lemma6,7,8) m k1 p 1 that fm¹zº has radius of convergence around 0 equal to 1/ k1 > 1 and has nite norm with respect to the Dirichlet inner product, hence fm 2 D. According to Proposition9 the norm of fm is given explicitly by m¹q−m − qmº k f k2 = k k ; (4) m D 4 where qk 2 ¹0; 1º is the (elliptic) nome of modulus k (see (61) in AppendixA), which is analytic for k in ˆ 1 the unit disk. Once properly normalized the family of functions ¹fmºm=1 provides an orthonormal basis of D, i.e. f ¹zº cos¹2πm¹v ¹zº + 1/4ºº − cos¹πm/2º D E ˆ ¹ º m k1 ˆ ˆ fm z B = q ; fn; fm = 1n=m : kfm kD m ¹ −m − mº D 4 qk qk The main technical result of this paper is the following. ≥ 2 π ¹αº Theorem 1. For `;p 1 and α 2 Z, let W`;p be the set of (possibly empty) simple diagonal walks w on Z2 n f¹0; 0ºg that start at ¹p; 0º, end on one of the axes at distance ` from the origin, and have full winding angle θw = α. ¹αº jw j ¹αº ¹ º Í ¹ º 2 ¹ º (i) Let W`;p t B 2 α t be the generating function of W`;p . For k = 4t 0; 1 xed, there w W`;p ¹αº D ¹ º1 exists a compact self-adjoint operator Yk on with eigenvectors fm m=1 such that ¹ º ¹ º ¹ º 2K¹kº 1 j j/ W α ¹tº = he ; Y α e i ; Y α f = qm α π f : (5) `;p ` k p D k m π m k m ¹α;I º ⊂ ¹αº (ii) Let W`;p W`;p be the subset of the aforementioned walks that have intermediate winding angles ⊂ w 2 j j − ¹α;I º¹ º in an interval I R, i.e.
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