On a connection between planar map combinatorics and lattice walks Timothy Budd

Transcription

1 Workshop on Large Random Structures in Two Dimensions, IHP, January 19th, 2017 On a connection between planar map combinatorics and lattice walks Timothy Budd IPhT, CEA-Saclay timothy.budd@cea.fr,

2 Introduction: Hyperbolic secant law The winding angle θ t of 2d Brownian motion satisfies Spitzer s law [Spitzer 58] 2θ t (d) log(t) Cauchy t

3 Introduction: Hyperbolic secant law The winding angle θ t of 2d Brownian motion satisfies Spitzer s law [Spitzer 58] 2θ t (d) log(t) Cauchy t

4 Introduction: Hyperbolic secant law The winding angle θ t of 2d Brownian motion satisfies Spitzer s law [Spitzer 58] [ ] 2θt t P (a, b) 1 b dx log(t) π 1 + x 2 a

5 Introduction: Hyperbolic secant law The winding angle θ t of 2d Brownian motion satisfies Spitzer s law [Spitzer 58] [ ] 2θt t P (a, b) 1 b dx log(t) π 1 + x 2 The winding angle θ n of 2d random walk satisfies hyperbolic secant law [Rudnick, Hu 87] [Bélisle 89] P [ 2θn (a, b) log(n) ] n 1 2 a b a ( πx sech 2 ) dx

6 Introduction: Hyperbolic secant law The winding angle θ t of 2d Brownian motion satisfies Spitzer s law [Spitzer 58] [ ] 2θt t P (a, b) 1 b dx log(t) π 1 + x 2 The winding angle θ n of 2d random walk satisfies hyperbolic secant law [Rudnick, Hu 87] [Bélisle 89] P [ 2θn (a, b) log(n) ] n 1 2 a b a ( πx sech 2 ) dx Surprising discrete analogue for SRW started at ( 1 2, 1 2 ): if n p 1 is geometric with parameter p, then for a, b Z: [ ] b θnp 1 2 P (a, b) = C p sech(πxt p ) π x=a+ 1 2

7 Introduction: Hyperbolic secant law The winding angle θ t of 2d Brownian motion satisfies Spitzer s law [Spitzer 58] [ ] 2θt t P (a, b) 1 b dx log(t) π 1 + x 2 a The winding angle θ n of 2d random walk satisfies hyperbolic secant law [Rudnick, Hu 87] [Bélisle 89] P [ 2θn (a, b) log(n) ] n 1 2 b a ( πx sech 2 ) dx ( ) Surprising discrete analogue for SRW started at ( 1 2, 1 2 ): if n p 1 is geometric with parameter p, then for a, b Z: [ ] b θnp 1 2 P (a, b) = C p sech(πxt p ) π T p 1 log(1 p) x=a+ 1 2 as p 1. Reproduces ( ).

8 Introduction: Gessel numbers In 2001 Ira Gessel conjectured the number of walks with 2n steps {N, S, SW, NE} in the quadrant starting and ending at 0 to be 16 n (5/6) n(1/2) n (2) n (5/3) n = 2, 11, 85, 782,...

9 Introduction: Gessel numbers In 2001 Ira Gessel conjectured the number of walks with 2n steps {N, S, SW, NE} in the quadrant starting and ending at 0 to be 16 n (5/6) n(1/2) n (2) n (5/3) n = 2, 11, 85, 782,... Turned out to be a notoriously difficult problem, but by now we have a computer-aided proof. [Kauers, Koutschan, Zeilberger, 08]... a human (complex-analytic) proof. [Bostan, Kurkova, Raschel, 13]... an elementary (algebraic) proof. [Bousquet-Mélou, 15]

10 Introduction: Gessel numbers In 2001 Ira Gessel conjectured the number of walks with 2n steps {N, S, SW, NE} in the quadrant starting and ending at 0 to be 16 n (5/6) n(1/2) n (2) n (5/3) n = 2, 11, 85, 782,... Turned out to be a notoriously difficult problem, but by now we have a computer-aided proof. [Kauers, Koutschan, Zeilberger, 08]... a human (complex-analytic) proof. [Bostan, Kurkova, Raschel, 13]... an elementary (algebraic) proof. [Bousquet-Mélou, 15] We will see that control of winding numbers provides an alternative route.

11 Introduction: planar maps Planar map = rooted planar graph embedded in R 2 up to homeomorphisms.

12 Introduction: planar maps Planar map = rooted planar graph embedded in R 2 up to homeomorphisms. Generating function of maps with fixed root face degree p: W (p) ({q i }) = q degree(f ) maps faces f

13 Introduction: planar maps Planar map = rooted planar graph embedded in R 2 up to homeomorphisms. Generating function of maps with fixed root face degree p: W (p) ({q i }) = q degree(f ) maps faces f Similarly, let W (p,0) be GF of maps with a marked vertex and W (p,l) for maps with a marked face of degree l. (Root face and marked face receive no weight!)

14 Introduction: planar maps Planar map = rooted planar graph embedded in R 2 up to homeomorphisms. Generating function of maps with fixed root face degree p: W (p) ({q i }) = q degree(f ) maps faces f Similarly, let W (p,0) be GF of maps with a marked vertex and W (p,l) for maps with a marked face of degree l. (Root face and marked face receive no weight!)

15 Relation between maps and walks? Classical result: for bipartite maps the GF with marked vertex takes a universal form (with ρ q a formal power series in q 2, q 4,...) ( ) 2p (ρq ) p W (2p,0) = p 4

16 Relation between maps and walks? Classical result: for bipartite maps the GF with marked vertex takes a universal form (with ρ q a formal power series in q 2, q 4,...) ( ) 2p (ρq ) p W (2p,0) = p 4 Same formula appears in GF s for lattice walks (2p, 0) (0, 0) that avoid negative half-axis (counted with factor t per step): not only staircase walks (ρ 4t 2 )...

17 Relation between maps and walks? Classical result: for bipartite maps the GF with marked vertex takes a universal form (with ρ q a formal power series in q 2, q 4,...) ( ) 2p (ρq ) p W (2p,0) = p 4 Same formula appears in GF s for lattice walks (2p, 0) (0, 0) that avoid negative half-axis (counted with factor t per step): not only staircase walks (ρ 4t 2 ) but whole class of walks on slit plane (ρ some power series in t). [Bousquet-Mélou, Schaeffer, 00]

18 Relation between maps and walks? Classical result: for bipartite maps the GF with marked vertex takes a universal form (with ρ q a formal power series in q 2, q 4,...) ( ) 2p (ρq ) p W (2p,0) = p 4 Same formula appears in GF s for lattice walks (2p, 0) (0, 0) that avoid negative half-axis (counted with factor t per step): not only staircase walks (ρ 4t 2 ) but whole class of walks on slit plane (ρ some power series in t). [Bousquet-Mélou, Schaeffer, 00] 1 in particular simple diagonal walks (ρ 1 16t 2 8t 2 1).

19 Relation between maps and walks? Continued. The GF for quasi-bipartite maps with a marked face has an equally universal form (see e.g. [Collet, Fusy, 12]) W (p,l) = 1 2 ( l p + l α(l)α(p) ρq ) (p+l)/2 p! α(l) := 4 p p 1 2! 2!

20 Relation between maps and walks? Continued. The GF for quasi-bipartite maps with a marked face has an equally universal form (see e.g. [Collet, Fusy, 12]) W (p,l) = 1 2 ( l p + l α(l)α(p) ρq ) (p+l)/2 p! α(l) := 4 p p 1 2! 2! Up to factor of two (and ρ t 2 8t 1) this also counts walks 2 on slit plane ending at ( l, 0). H (p,l) (t) = 2 ( ) (p+l)/2 2 ρ(t) l p + l α(l)α(p) 4

21 Relation between maps and walks? Continued. The GF for quasi-bipartite maps with a marked face has an equally universal form (see e.g. [Collet, Fusy, 12]) W (p,l) = 1 l 2 ( p + l α(l)α(p) ρq ) (p+l)/2 α(l) := 4 p! p p 1 2! 2! Up to factor of two (and ρ t 2 8t 1) this also counts walks 2 on slit plane ending at ( l, 0). H (p,l) (t) = 2 ( ) (p+l)/2 2 ρ(t) l p + l α(l)α(p) 4

22 Relation between maps and walks? Continued. The GF for quasi-bipartite maps with a marked face has an equally universal form (see e.g. [Collet, Fusy, 12]) W (p,l) = 1 l 2 ( p + l α(l)α(p) ρq ) (p+l)/2 α(l) := 4 p! p p 1 2! 2! Up to factor of two (and ρ t 2 8t 1) this also counts walks 2 on slit plane ending at ( l, 0). H (p,l) (t) = 2 ( ) (p+l)/2 2 ρ(t) p l p + l α(l)α(p) =: 4 l (H) pl

23 Relation between maps and walks? Continued. The GF for quasi-bipartite maps with a marked face has an equally universal form (see e.g. [Collet, Fusy, 12]) W (p,l) = 1 l 2 ( p + l α(l)α(p) ρq ) (p+l)/2 α(l) := 4 p! p p 1 2! 2! Up to factor of two (and ρ t 2 8t 1) this also counts walks 2 on slit plane ending at ( l, 0). Coincidence? H (p,l) (t) = 2 ( ) (p+l)/2 2 ρ(t) p l p + l α(l)α(p) =: 4 l (H) pl

24 A bijective explanation Proposition For any step set S { 1, 0, 1, 2,...} { 1, 0, 1}, there is a 2-to-1 map Φ (p,l) : {S-walks (p, 0),, ( l, 0) on slit plane} { } S-walk-decorated maps with root face degree p and marked face degree l

25 A bijective explanation Proposition For any step set S { 1, 0, 1, 2,...} { 1, 0, 1}, there is a 2-to-1 map Φ (p,l) : {S-walks (p, 0),, ( l, 0) on slit plane} { } S-walk-decorated maps with root face degree p and marked face degree l A S-walk-decorated map is a rooted planar map with a marked face together with...

26 A bijective explanation Proposition For any step set S { 1, 0, 1, 2,...} { 1, 0, 1}, there is a 2-to-1 map Φ (p,l) : {S-walks (p, 0),, ( l, 0) on slit plane} { } S-walk-decorated maps with root face degree p and marked face degree l A S-walk-decorated map is a rooted planar map with a marked face together with... for each face (except root or marked) of degree k an excursion (0, 0),..., (k 2, 0) above or below x-axis.

27 A bijective explanation Proposition For any step set S { 1, 0, 1, 2,...} { 1, 0, 1}, there is a 2-to-1 map Φ (p,l) : {S-walks (p, 0),, ( l, 0) on slit plane} { } S-walk-decorated maps with root face degree p and marked face degree l A S-walk-decorated map is a rooted planar map with a marked face together with... for each face (except root or marked) of degree k an excursion (0, 0),..., (k 2, 0) above or below x-axis. for each vertex an excursion (0, 0),..., ( 2, 0) above x-axis

28 A bijective explanation Proposition For any step set S { 1, 0, 1, 2,...} { 1, 0, 1}, there is a 2-to-1 map Φ (p,l) : {S-walks (p, 0),, ( l, 0) on slit plane} { } S-walk-decorated maps with root face degree p and marked face degree l A S-walk-decorated map is a rooted planar map with a marked face together with... for each face (except root or marked) of degree k an excursion (0, 0),..., (k 2, 0) above or below x-axis. for each vertex an excursion (0, 0),..., ( 2, 0) above x-axis Substituting in 2W ( (p,l) ({q i }) ) ( the GF ) k 2 2 q k leads to H (p,l) (t) (up to ( ) p+l ).

29

30

31

32 From walks to (rigid) loop-decorated maps

33 From walks to (rigid) loop-decorated maps

34 From walks to (rigid) loop-decorated maps

35 From walks to (rigid) loop-decorated maps

36 From walks to (rigid) loop-decorated maps

37 From walks to (rigid) loop-decorated maps

38 From walks to (rigid) loop-decorated maps

39 From walks to (rigid) loop-decorated maps

40 From walks to (rigid) loop-decorated maps

41 From walks to (rigid) loop-decorated maps Such walks from (p, 0) to (±l, 0) with winding angle θ w have GF G (p,l) b := w ( e t w ibπ + e ibπ ) N e ibθw = H (p,k1) H (k1,k2) H (k N 1,l) 2 N=1 k 1,...,k N 1 1

42 From walks to (rigid) loop-decorated maps Such walks from (p, 0) to (±l, 0) with winding angle θ w have GF G (p,l) b := w t w e ibθw = cos N (πb) H (p,k1) H (k1,k2) H (k N 1,l) N=1 k 1,...,k N 1 1

43 From walks to (rigid) loop-decorated maps Such walks from (p, 0) to (±l, 0) with winding angle θ w have GF G (p,l) b := w t w e ibθw = N=1 cos N (πb) p l (HN ) pl

44 From walks to (rigid) loop-decorated maps Such walks from (p, 0) to (±l, 0) with winding angle θ w have GF G (p,l) b := w t w e ibθw = p l ( cos(πb)h I cos(πb)h ) pl

45 From walks to (rigid) loop-decorated maps Such walks from (p, 0) to (±l, 0) with winding angle θ w have GF G (p,l) b := w t w e ibθw = p l ( ) cos(πb)h I cos(πb)h pl But this also enumerates planar maps decorated with rigid loops carrying a weight n := 2 cos(πb) each (and a redundant overall factor of n).

46 Planar maps coupled to a rigid O(n) loop model Rigid O(n) model: a planar map + disjoint loops, that intersect solely quadrangles through opposite sides. Enumerated with weight n #loops g #loop faces regular faces q degree An exact solution of a closely related model was first obtained by [Eynard, Kristjansen, 95] in terms of elliptic functions.

47 Planar maps coupled to a rigid O(n) loop model Rigid O(n) model: a planar map + disjoint loops, that intersect solely quadrangles through opposite sides. Enumerated with weight n #loops g #loop faces regular faces q degree An exact solution of a closely related model was first obtained by [Eynard, Kristjansen, 95] in terms of elliptic functions. Made more precise in [Borot, Eynard, 09], and in [Borot, Bouttier, Guitter, 11] for this rigid setting.

48 Planar maps coupled to a rigid O(n) loop model Rigid O(n) model: a planar map + disjoint loops, that intersect solely quadrangles through opposite sides. Enumerated with weight n #loops g #loop faces regular faces q degree An exact solution of a closely related model was first obtained by [Eynard, Kristjansen, 95] in terms of elliptic functions. Made more precise in [Borot, Eynard, 09], and in [Borot, Bouttier, Guitter, 11] for this rigid setting. Recently in [Borot, Bouttier, Duplantier, 16] (in slightly different setting) exact statistics for the nesting of loops was obtained, i.e. distribution of # loops surrounding a marked vertex/face.

49 Planar maps coupled to a rigid O(n) loop model Rigid O(n) model: a planar map + disjoint loops, that intersect solely quadrangles through opposite sides. Enumerated with weight n #loops g #loop faces regular faces q degree An exact solution of a closely related model was first obtained by [Eynard, Kristjansen, 95] in terms of elliptic functions. Made more precise in [Borot, Eynard, 09], and in [Borot, Bouttier, Guitter, 11] for this rigid setting. Recently in [Borot, Bouttier, Duplantier, 16] (in slightly different setting) exact statistics for the nesting of loops was obtained, i.e. distribution of # loops surrounding a marked vertex/face. Importantly: suppressing loops that do not surround mark affects GF s only through renormalization of q.

50 Adapting GF from [Borot, Bouttier, Duplantier, 16], setting n = 2 cos(πb) and computing a series representation: G b (x 1, x 2 ; t) := = 4 m=1 p,l 1 x p 1 x l 2 G (p,l) b 2 cos(πb) cos(2πm v(x 2 )) x 1 x 1 cos(2πm v(x 1 )) q m + q m 2 cos(πb) m(q m q m ) where q = q(4t) is elliptic nome of modulus 4t and v(x) := cd 1 ( x/ ρ, ρ)/(4k(ρ)), ρ(t) = t 2 8t 2 1

51 Adapting GF from [Borot, Bouttier, Duplantier, 16], setting n = 2 cos(πb) and computing a series representation: G b (x 1, x 2 ; t) := x p 1 x 2 l G (p,l) b = ( ) x p p cos(πb)h 1 x 2 l l I cos(πb)h p,l 1 p,l 1 pl 2 cos(πb) cos(2πm v(x 2 )) x 1 x = 4 1 cos(2πm v(x 1 )) q m + q m 2 cos(πb) m(q m q m ) m=1 where q = q(4t) is elliptic nome of modulus 4t and v(x) := cd 1 ( x/ ρ, ρ)/(4k(ρ)), ρ(t) = t 2 8t 2 1

52 Adapting GF from [Borot, Bouttier, Duplantier, 16], setting n = 2 cos(πb) and computing a series representation: G b (x 1, x 2 ; t) := x p 1 x 2 l G (p,l) b = ( ) x p p cos(πb)h 1 x 2 l l I cos(πb)h p,l 1 p,l 1 pl 2 cos(πb) cos(2πm v(x 2 )) x 1 x = 4 1 cos(2πm v(x 1 )) q m + q m 2 cos(πb) m(q m q m ) m=1 where q = q(4t) is elliptic nome of modulus 4t and v(x) := cd 1 ( x/ ρ, ρ)/(4k(ρ)), ρ(t) = t 2 8t 2 1 Proposition (Diagonalization of H) H = U T Λ q U in the sense of operators on l 2 (R) with ( 2 Λ q = diag q m + q m ), U mp = m 1 4p m(q m q m ) [x p ] cos(2πm v(x))

53 Application 1: hyperbolic secant law Recall p l (HN ) pl enumerates walks (p, 0) (±l, 0) that alternate between half-axes N times.

54 Application 1: hyperbolic secant law 1 Recall l (HN ) 1l enumerates walks (1, 0) (±l, 0) that alternate between half-axes N times.

55 Application 1: hyperbolic secant law 1 Recall l (HN ) 1l enumerates walks (1, 0) (±l, 0) that alternate between half-axes N times. l 1 1 l (H N ) 1,l Then 1 1 4t enumerates all walks alternating N times.

56 Application 1: hyperbolic secant law 1 Recall l (HN ) 1l enumerates walks (1, 0) (±l, 0) that alternate between half-axes N times. l 1 1 l (H N H N+1 ) 1,l Then 1 1 4t enumerates all walks alternating exactly N times.

57 Application 1: hyperbolic secant law 1 Recall l (HN ) 1l enumerates walks (1, 0) (±l, 0) that alternate between half-axes N times. l 1 1 l (H N H N+1 ) 1,l Then 1 1 4t enumerates all walks alternating exactly N times. t w e iπb( θw π ) = w 4t cos(πb/2) 1 4t N 0 cos N (πb) l 1 1 l (H N H N+1 ) 1,l

58 Application 1: hyperbolic secant law 1 Recall l (HN ) 1l enumerates walks (1, 0) (±l, 0) that alternate between half-axes N times. l 1 1 l (H N H N+1 ) 1,l Then 1 1 4t enumerates all walks alternating exactly N times. t w e iπb( θw π ) = w = 1 π 1 4t 2K(4t) 4t cos(πb/2) 1 4t k= N 0 cos N (πb) l 1 2e iπb(k+ 1 2 ) q k q k 1 2 = 1 l (H N H N+1 ) 1,l cn(b K(4t), 4t) 1 4t

59 Application 1: hyperbolic secant law 1 Recall l (HN ) 1l enumerates walks (1, 0) (±l, 0) that alternate between half-axes N times. l 1 1 l (H N H N+1 ) 1,l Then 1 1 4t enumerates all walks alternating exactly N times. t w e iπb( θw π ) = w = 1 π 1 4t 2K(4t) 4t cos(πb/2) 1 4t k= N 0 cos N (πb) l 1 2e iπb(k+ 1 2 ) q k q k 1 2 = 1 l (H N H N+1 ) 1,l cn(b K(4t), 4t) 1 4t Theorem (Winding angle of SRW on Z 2 around ( 1 2, 1 2 )) If n p 1 is a geometric RV with parameter 0 < p < 1 then P [ kπ < θ np < (k + 1)π ] sech(π(k = )T ) K( 1 p2 ) k Z sech(π(k + 1 T = 2 )T ), K(p)

60 Refinement: increase winding angle resolution Up to now: decomposed walk into sequence of walks on slit plane.

61 Refinement: increase winding angle resolution Up to now: decomposed walk into sequence of walks on slit plane.

62 Refinement: increase winding angle resolution Up to now: decomposed walk into sequence of walks on slit plane. Why not decompose into walks on half plane?

63 Refinement: increase winding angle resolution Up to now: decomposed walk into sequence of walks on slit plane. Why not decompose into walks on half plane?

64 Refinement: increase winding angle resolution Up to now: decomposed walk into sequence of walks on slit plane. Why not decompose into walks on half plane?

65 Refinement: increase winding angle resolution Up to now: decomposed walk into sequence of walks on slit plane. Why not decompose into walks on half plane?

66 Refinement: increase winding angle resolution Up to now: decomposed walk into sequence of walks on slit plane. Why not decompose into walks on half plane? Denote GF for half-plane walks (p, 0),... (0, ±l) by p l J pl. Then H = 1 2 J 2 (I + H), J = 2H I + H

67 Refinement: increase winding angle resolution Up to now: decomposed walk into sequence of walks on slit plane. Why not decompose into walks on half plane? Denote GF for half-plane walks (p, 0),... (0, ±l) by p l J pl. Then H = 1 2 J 2 (I + H), J = 2H I + H 2 Hence J has same eigenmodes as H but eigenvalues are q m/2 +q m/2 2 instead of q m +q. Such an operation q q on elliptic functions m are well-known as Landen transformations.

68 Winding angle of excursions Wish to enumerate excursions from origin by length and winding angle: F (t, b) := w t w ib θw e = 4t 2 + (12 + 4e ib π 2 + 4e ib π 2 )t

69 Winding angle of excursions Wish to enumerate excursions from origin by length and winding angle: F (t, b) := w t w ib θw e = 4t 2 + (12 + 4e ib π 2 + 4e ib π 2 )t Flip last step away from last axis intersection, and first step oppositely.

70 Winding angle of excursions Wish to enumerate excursions from origin by length and winding angle: F (t, b) := w t w ib θw e = 4t 2 + (12 + 4e ib π 2 + 4e ib π 2 )t Flip last step away from last axis intersection, and first step oppositely. θ w now measures angle to penultimate axis intersection.

71 Winding angle of excursions Wish to enumerate excursions from origin by length and winding angle: F (t, b) := w t w ib θw e = 4t 2 + (12 + 4e ib π 2 + 4e ib π 2 )t Flip last step away from last axis intersection, and first step oppositely. θ w now measures angle to penultimate axis intersection. This maps excursions 4-to-2 onto sequence of half-plane walks (2, 0), (1, ±1),... (±1, ±1), (0/±2, 0/±2).

72 Winding angle of excursions Wish to enumerate excursions from origin by length and winding angle: F (t, b) := w t w ib θw e = 4t 2 + (12 + 4e ib π 2 + 4e ib π 2 )t Flip last step away from last axis intersection, and first step oppositely. θ w now measures angle to penultimate axis intersection. This maps excursions 4-to-2 onto sequence of half-plane walks (2, 0), (1, ±1),... (±1, ±1), (0/±2, 0/±2). Enumerated by F (t, b) = 2 [ ( ( cos πb )) N 1 2 (J N ) 22 N 1 ]

73 Winding angle of excursions Wish to enumerate excursions from origin by length and winding angle: F (t, b) := w t w ib θw e = 4t 2 + (12 + 4e ib π 2 + 4e ib π 2 )t Flip last step away from last axis intersection, and first step oppositely. θ w now measures angle to penultimate axis intersection. This maps excursions 4-to-2 onto sequence of half-plane walks (2, 0), (1, ±1),... (±1, ±1), (0/±2, 0/±2). Enumerated by F (t, b) = 2 [ ( ( cos πb )) N 1 2 (J N ) 22 N 1 ]

74 Winding angle of excursions Wish to enumerate excursions from origin by length and winding angle: F (t, b) := w t w ib θw e = 4t 2 + (12 + 4e ib π 2 + 4e ib π 2 )t Flip last step away from last axis intersection, and first step oppositely. θ w now measures angle to penultimate axis intersection. This maps excursions 4-to-2 onto sequence of half-plane walks (2, 0), (1, ±1),... (±1, ±1), (0/±2, 0/±2). Enumerated by F (t, b) = 2 [ ( ( cos πb )) ] N 1 2 (J N 4 ) 22 2 (J N 6 ) (J N ) 62 N 1

75 Winding angle of excursions Wish to enumerate excursions from origin by length and winding angle: F (t, b) := w t w ib θw e = 4t 2 + (12 + 4e ib π 2 + 4e ib π 2 )t Flip last step away from last axis intersection, and first step oppositely. θ w now measures angle to penultimate axis intersection. This maps excursions 4-to-2 onto sequence of half-plane walks (2, 0), (1, ±1),... (±1, ±1), (0/±2, 0/±2). Enumerated by F (t, b) = 2 ( ( cos πb )) N 1 2 ( 1) p+l p l (J N ) 2p,2l N 1 p,l 0

76 Winding angle of excursions Wish to enumerate excursions from origin by length and winding angle: F (t, b) := w t w ib θw e = 4t 2 + (12 + 4e ib π 2 + 4e ib π 2 )t Flip last step away from last axis intersection, and first step oppositely. θ w now measures angle to penultimate axis intersection. This maps excursions 4-to-2 onto sequence of half-plane walks (2, 0), (1, ±1),... (±1, ±1), (0/±2, 0/±2). Enumerated by F (t, b) = 2 ( ( cos πb )) N 1 2 ( 1) p+l p l (J N ) 2p,2l N 1 = sec ( πb 2 p,l 0 ) [ 1 π tan ( πb 4 2K(4t) ) θ 1 ( πb 4, q) θ 1 ( πb 4, q) ]

77 Application 2: walks in cones Theorem (Excursions in the nπ 4 -cone.) For integers m n < p < m < n the GF for simple walks (0, 0), (1, 0),..., (0, 0) with winding angle pπ 2 staying strictly inside angular region ( p+m n 4 π, p+m 4 π) is F n,m,p (t) := 1 n 1 (e 2iπ ( ) pk n e 2iπ mk n ) F t, 4k 4n n k=1

78 Application 2: walks in cones Theorem (Excursions in the nπ 4 -cone.) For integers m n < p < m < n the GF for simple walks (0, 0), (1, 0),..., (0, 0) with winding angle pπ 2 staying strictly inside angular region ( p+m n 4 π, p+m 4 π) is F n,m,p (t) := 1 n 1 (e 2iπ ( ) pk n e 2iπ mk n ) F t, 4k 4n n k=1

79 Application 2: walks in cones Theorem (Excursions in the nπ 4 -cone.) For integers m n < p < m < n the GF for simple walks (0, 0), (1, 0),..., (0, 0) with winding angle pπ 2 staying strictly inside angular region ( p+m n 4 π, p+m 4 π) is F n,m,p (t) := 1 n 1 (e 2iπ ( ) pk n e 2iπ mk n ) F t, 4k 4n n k=1

80 Application 2: walks in cones Theorem (Excursions in the nπ 4 -cone.) For integers m n < p < m < n the GF for simple walks (0, 0), (1, 0),..., (0, 0) with winding angle pπ 2 staying strictly inside angular region ( p+m n 4 π, p+m 4 π) is F n,m,p (t) := 1 n 1 (e 2iπ ( ) pk n e 2iπ mk n ) F t, 4k 4n n k=1

81 Application 2: walks in cones Theorem (Excursions in the nπ 4 -cone.) For integers m n < p < m < n the GF for simple walks (0, 0), (1, 0),..., (0, 0) with winding angle pπ 2 staying strictly inside angular region ( p+m n 4 π, p+m 4 π) is F n,m,p (t) := 1 n 1 (e 2iπ ( ) pk n e 2iπ mk n ) F t, 4k 4n n k=1 Direct consequence of the reflection principle.

82 Application 2: walks in cones Theorem (Excursions in the nπ 4 -cone.) For integers m n < p < m < n the GF for simple walks (0, 0), (1, 0),..., (0, 0) with winding angle pπ 2 staying strictly inside angular region ( p+m n 4 π, p+m 4 π) is F n,m,p (t) := 1 n 1 (e 2iπ ( ) pk n e 2iπ mk n ) F t, 4k 4n n k=1 Direct consequence of the reflection principle.

83 Application 2: walks in cones (Gessel case) Special case: (n, m, p) = (3, 2, 0)

84 Application 2: walks in cones (Gessel case) Special case: (n, m, p) = (3, 2, 0)

85 Application 2: walks in cones (Gessel case) Special case: (n, m, p) = (3, 2, 0) Gessel-type walks returning to origin in quadrant are enumerated by 1 t 2 F 3,2,0(t) = 1 4t 2 F (t, 4 3 )

86 Application 2: walks in cones (Gessel case) Special case: (n, m, p) = (3, 2, 0) Gessel-type walks returning to origin in quadrant are enumerated by 1 t 2 F 3,2,0(t) = 1 4t 2 F (t, 4 3 ) [ = 1 3π θ 1 ( π 3, ] q) 2t 2 2K(4t) θ 1 ( π 3, q) 1 = 1 + 2t t t 6 + which is [OEIS sequence A135404]

87 Application 2: walks in cones (Gessel case) Special case: (n, m, p) = (3, 2, 0) Gessel-type walks returning to origin in quadrant are enumerated by 1 t 2 F 3,2,0(t) = 1 4t 2 F (t, 4 3 ) [ = 1 3π θ 1 ( π 3, ] q) 2t 2 2K(4t) θ 1 ( π 3, q) 1 = 1 + 2t t t 6 + which is [OEIS sequence A135404] But not obvious that this reproduces the known GF [ ] t 2n 16 n (5/6) n(1/2) n = 1 ( ) (2) n (5/3) n 2t 2 2F 1 1 2, 1 6 ; 2 3 ; (4t)2 1, n=0 nor that it is algebraic [Bostan, Kauers, 09].

88 Further questions Which generating functions are algebraic? Other walks with small steps? Why are some of the generating functions biperiodic and other ones only quasi-biperiodic? Finally, here is an interpretation of the nome q as function of the elliptic modulus k. Why is it so simple? [ SRW on Z q(k) = lim P 2 reaches winding angle nπ n before geometric time with parameter k ] 1/n