Counting lattice walks by winding angle Sminaire de combinatoire - PowerPoint PPT Presentation

K REWERAS WALKS BY WINDING NUMBER The model: Count walks starting at by end point. × e iα ( s − 1 ) × e − iα ( s ) This example contributes t 5 xy 3 . � t | p | x x ( p ) y y ( p ) e i α n ( p ) Definition: Q ( t , α, x , y ) ≡ Q ( x , y ) = paths p Note: Q ( 0 , 0 ) = E ( t , e i α ) Counting lattice walks by winding angle Andrew Elvey Price

K REWERAS WALKS BY WINDING NUMBER The model: Count walks starting at by end point. × e iα ( s − 1 ) × e − iα ( s ) This example contributes t 6 xy 2 . � t | p | x x ( p ) y y ( p ) e i α n ( p ) Definition: Q ( t , α, x , y ) ≡ Q ( x , y ) = paths p Note: Q ( 0 , 0 ) = E ( t , e i α ) Counting lattice walks by winding angle Andrew Elvey Price

K REWERAS WALKS BY WINDING NUMBER The model: Count walks starting at by end point. × e iα ( s − 1 ) × e − iα ( s ) This example contributes t 7 xy . � t | p | x x ( p ) y y ( p ) e i α n ( p ) Definition: Q ( t , α, x , y ) ≡ Q ( x , y ) = paths p Note: Q ( 0 , 0 ) = E ( t , e i α ) Counting lattice walks by winding angle Andrew Elvey Price

K REWERAS WALKS BY WINDING NUMBER The model: Count walks starting at by end point. × e iα ( s − 1 ) × e − iα ( s ) This example contributes t 8 x . � t | p | x x ( p ) y y ( p ) e i α n ( p ) Definition: Q ( t , α, x , y ) ≡ Q ( x , y ) = paths p Note: Q ( 0 , 0 ) = E ( t , e i α ) Counting lattice walks by winding angle Andrew Elvey Price

K REWERAS WALKS BY WINDING NUMBER The model: Count walks starting at by end point. × e iα ( s − 1 ) × e − iα ( s ) This example contributes t 9 y 2 e − i α . � t | p | x x ( p ) y y ( p ) e i α n ( p ) Definition: Q ( t , α, x , y ) ≡ Q ( x , y ) = paths p Note: Q ( 0 , 0 ) = E ( t , e i α ) Counting lattice walks by winding angle Andrew Elvey Price

K REWERAS WALKS BY WINDING NUMBER The model: Count walks starting at by end point. × e iα ( s − 1 ) × e − iα ( s ) This example contributes t 10 xy 3 e − i α . � t | p | x x ( p ) y y ( p ) e i α n ( p ) Definition: Q ( t , α, x , y ) ≡ Q ( x , y ) = paths p Note: Q ( 0 , 0 ) = E ( t , e i α ) Counting lattice walks by winding angle Andrew Elvey Price

F UNCTIONAL EQUATION Recursion → functional equation: separate by type of final step. Q ( x, y ) = 1 + xytQ ( x, y ) + e iα tQ (0 , x ) + (Final step goes through left wall) t x ( Q ( x, y ) − Q (0 , y )) + e − iα tyQ ( y, 0) + (Final step goes through bottom wall) t y ( Q ( x, y ) − Q ( x, 0)) Counting lattice walks by winding angle Andrew Elvey Price

K REWERAS WALKS BY WINDING NUMBER The model: Count walks starting at the red point by end point. × e iα ( s − 1 ) × e − iα ( s ) � t | p | x x ( p ) y y ( p ) e i α n ( p ) . Definition: Q ( t , α, x , y ) ≡ Q ( x , y ) = paths p Characterised by: Q ( x , y ) = 1 + txyQ ( x , y ) + tQ ( x , y ) − Q ( 0 , y ) + tQ ( x , y ) − Q ( x , 0 ) x y + e i α tQ ( 0 , x ) + e − i α tyQ ( y , 0 ) . Counting lattice walks by winding angle Andrew Elvey Price

Part 2: Solution (using theta functions) Counting lattice walks by winding angle Andrew Elvey Price

S OLUTION TO K REWERAS WALKS BY WINDING NUMBER Equation to solve: Q ( x , y ) = 1 + txyQ ( x , y ) + tQ ( x , y ) − Q ( 0 , y ) + tQ ( x , y ) − Q ( x , 0 ) x y + e i α tQ ( 0 , x ) + e − i α tyQ ( y , 0 ) . Counting lattice walks by winding angle Andrew Elvey Price

S OLUTION TO K REWERAS WALKS BY WINDING NUMBER Equation to solve: Q ( x , y ) = 1 + txyQ ( x , y ) + tQ ( x , y ) − Q ( 0 , y ) + tQ ( x , y ) − Q ( x , 0 ) x y + e i α tQ ( 0 , x ) + e − i α tyQ ( y , 0 ) . Solution: Step 1: Fix t ∈ [ 0 , 1 / 3 ) , α ∈ R . All series converge for | x | , | y | < 1. Counting lattice walks by winding angle Andrew Elvey Price

S OLUTION TO K REWERAS WALKS BY WINDING NUMBER Equation to solve: Q ( x , y ) = 1 + txyQ ( x , y ) + tQ ( x , y ) − Q ( 0 , y ) + tQ ( x , y ) − Q ( x , 0 ) x y + e i α tQ ( 0 , x ) + e − i α tyQ ( y , 0 ) . Solution: Step 1: Fix t ∈ [ 0 , 1 / 3 ) , α ∈ R . All series converge for | x | , | y | < 1. Step 2: Write equation as K ( x , y ) Q ( x , y ) = R ( x , y ) , where K ( x , y ) = 1 − txy − t / y − t / x R ( x , y ) = 1 − t xQ ( 0 , y ) − t yQ ( x , 0 ) + e i α tQ ( 0 , x ) + e − i α tyQ ( y , 0 ) . Counting lattice walks by winding angle Andrew Elvey Price

S OLUTION TO K REWERAS WALKS BY WINDING NUMBER Equation to solve: Q ( x , y ) = 1 + txyQ ( x , y ) + tQ ( x , y ) − Q ( 0 , y ) + tQ ( x , y ) − Q ( x , 0 ) x y + e i α tQ ( 0 , x ) + e − i α tyQ ( y , 0 ) . Solution: Step 1: Fix t ∈ [ 0 , 1 / 3 ) , α ∈ R . All series converge for | x | , | y | < 1. Step 2: Write equation as K ( x , y ) Q ( x , y ) = R ( x , y ) , where K ( x , y ) = 1 − txy − t / y − t / x R ( x , y ) = 1 − t xQ ( 0 , y ) − t yQ ( x , 0 ) + e i α tQ ( 0 , x ) + e − i α tyQ ( y , 0 ) . Step 3: Consider the curve K ( x , y ) = 0 (Then R ( x , y ) = 0). Counting lattice walks by winding angle Andrew Elvey Price

S OLUTION TO K REWERAS WALKS BY WINDING NUMBER Equation to solve: Q ( x , y ) = 1 + txyQ ( x , y ) + tQ ( x , y ) − Q ( 0 , y ) + tQ ( x , y ) − Q ( x , 0 ) x y + e i α tQ ( 0 , x ) + e − i α tyQ ( y , 0 ) . Solution: Step 1: Fix t ∈ [ 0 , 1 / 3 ) , α ∈ R . All series converge for | x | , | y | < 1. Step 2: Write equation as K ( x , y ) Q ( x , y ) = R ( x , y ) , where K ( x , y ) = 1 − txy − t / y − t / x R ( x , y ) = 1 − t xQ ( 0 , y ) − t yQ ( x , 0 ) + e i α tQ ( 0 , x ) + e − i α tyQ ( y , 0 ) . Step 3: Consider the curve K ( x , y ) = 0 (Then R ( x , y ) = 0). Parameterisation involves the Jacobi theta function ϑ ( z , τ ) . So far: Similar to elliptic approaches to quadrant models [Bernardi, Bousquet-Mélou, Fayolle, Iasnogorodski, Kurkova, Malyshev, Raschel, Trotignon] Counting lattice walks by winding angle Andrew Elvey Price

J ACOBI THETA FUNCTION ϑ ( z , τ ) Definition: For τ, z ∈ C , im ( τ ) > 0, ∞ 2 i πτ +( 2 n + 1 ) iz ( − 1 ) n e ( 2 n + 1 2 ) � ϑ ( z , τ ) = n = −∞ Useful facts (for fixed τ ): ϑ ( z + π, τ ) = − ϑ ( z , τ ) ϑ ( z + πτ, τ ) = − e − 2 iz − i πτ ϑ ( z , τ ) Counting lattice walks by winding angle Andrew Elvey Price

P ARAMETERISATION OF K ( x , y ) = 0 USING ϑ ( z , τ ) Definition: For τ, z ∈ C , im ( τ ) > 0, ∞ 2 i πτ +( 2 n + 1 ) iz ( − 1 ) n e ( 2 n + 1 2 ) � ϑ ( z , τ ) = n = −∞ Useful facts (for fixed τ ): ϑ ( z + π, τ ) = − ϑ ( z , τ ) ϑ ( z + πτ, τ ) = − e − 2 iz − i πτ ϑ ( z , τ ) Parameterisation: The curve K ( x , y ) := 1 − txy − t / y − t / x = 0 is parameterised by X ( z ) = e − 4 πτ i 3 ϑ ( z , 3 τ ) ϑ ( z − πτ, 3 τ ) and Y ( z ) = X ( z + πτ ) , ϑ ( z + πτ, 3 τ ) ϑ ( z − 2 πτ, 3 τ ) ϑ ′ ( 0 , 3 τ ) where τ is determined by t = e − πτ i 4 i ϑ ( πτ, 3 τ ) + 6 ϑ ′ ( πτ, 3 τ ) . 3 Counting lattice walks by winding angle Andrew Elvey Price

S OLUTION TO K REWERAS WALKS BY WINDING NUMBER Equation to solve: K ( x , y ) Q ( x , y ) = R ( x , y ) , where K ( x , y ) = 1 − txy − t / y − t / x , R ( x , y ) = 1 − t xQ ( 0 , y ) − t yQ ( x , 0 ) + e i α tQ ( 0 , x ) + e − i α tyQ ( y , 0 ) . Counting lattice walks by winding angle Andrew Elvey Price

S OLUTION TO K REWERAS WALKS BY WINDING NUMBER Equation to solve: K ( x , y ) Q ( x , y ) = R ( x , y ) , where K ( x , y ) = 1 − txy − t / y − t / x , R ( x , y ) = 1 − t xQ ( 0 , y ) − t yQ ( x , 0 ) + e i α tQ ( 0 , x ) + e − i α tyQ ( y , 0 ) . Define X ( z ) = e − 4 πτ i 3 ϑ ( z , 3 τ ) ϑ ( z − πτ, 3 τ ) ϑ ( z + πτ, 3 τ ) ϑ ( z − 2 πτ, 3 τ ) . Then K ( X ( z ) , X ( z + πτ )) = 0. Counting lattice walks by winding angle Andrew Elvey Price

S OLUTION TO K REWERAS WALKS BY WINDING NUMBER Equation to solve: K ( x , y ) Q ( x , y ) = R ( x , y ) , where K ( x , y ) = 1 − txy − t / y − t / x , R ( x , y ) = 1 − t xQ ( 0 , y ) − t yQ ( x , 0 ) + e i α tQ ( 0 , x ) + e − i α tyQ ( y , 0 ) . Define X ( z ) = e − 4 πτ i 3 ϑ ( z , 3 τ ) ϑ ( z − πτ, 3 τ ) ϑ ( z + πτ, 3 τ ) ϑ ( z − 2 πτ, 3 τ ) . Then K ( X ( z ) , X ( z + πτ )) = 0. Hence R ( X ( z ) , X ( z + πτ )) = 0 (assuming | X ( z ) | ≤ 1 and | X ( z + πτ ) | ≤ 1). Counting lattice walks by winding angle Andrew Elvey Price

S OLUTION TO K REWERAS WALKS BY WINDING NUMBER Equation to solve: K ( x , y ) Q ( x , y ) = R ( x , y ) , where K ( x , y ) = 1 − txy − t / y − t / x , R ( x , y ) = 1 − t xQ ( 0 , y ) − t yQ ( x , 0 ) + e i α tQ ( 0 , x ) + e − i α tyQ ( y , 0 ) . Define X ( z ) = e − 4 πτ i 3 ϑ ( z , 3 τ ) ϑ ( z − πτ, 3 τ ) ϑ ( z + πτ, 3 τ ) ϑ ( z − 2 πτ, 3 τ ) . Then K ( X ( z ) , X ( z + πτ )) = 0. Hence R ( X ( z ) , X ( z + πτ )) = 0 (assuming | X ( z ) | ≤ 1 and | X ( z + πτ ) | ≤ 1). New equation to solve: R ( X ( z ) , X ( z + πτ )) = 0 , Counting lattice walks by winding angle Andrew Elvey Price

S OLUTION TO K REWERAS WALKS BY WINDING NUMBER Equation to solve: K ( x , y ) Q ( x , y ) = R ( x , y ) , where K ( x , y ) = 1 − txy − t / y − t / x , R ( x , y ) = 1 − t xQ ( 0 , y ) − t yQ ( x , 0 ) + e i α tQ ( 0 , x ) + e − i α tyQ ( y , 0 ) . Define X ( z ) = e − 4 πτ i 3 ϑ ( z , 3 τ ) ϑ ( z − πτ, 3 τ ) ϑ ( z + πτ, 3 τ ) ϑ ( z − 2 πτ, 3 τ ) . Then K ( X ( z ) , X ( z + πτ )) = 0. Hence R ( X ( z ) , X ( z + πτ )) = 0 (assuming | X ( z ) | ≤ 1 and | X ( z + πτ ) | ≤ 1). New equation to solve: R ( X ( z ) , X ( z + πτ )) = 0 , Counting lattice walks by winding angle Andrew Elvey Price

S OLUTION TO K REWERAS WALKS BY WINDING NUMBER � � � � 1 � � 0 , 1 Plot of z : | X ( z ) | ∈ , 3 , 1 , ( 1 , 3 ) , ( 3 , 9 ) , ( 9 , ∞ ] . 3 2 πτ πτ Ω 0 π 3 π 2 π − π − πτ − 2 πτ For z ∈ Ω , | X ( z ) | < 1 ⇒ Q ( X ( z ) , 0 ) and Q ( 0 , X ( z )) are well defined. Counting lattice walks by winding angle Andrew Elvey Price

S OLUTION TO K REWERAS WALKS BY WINDING NUMBER � � � � 1 � � 0 , 1 Plot of z : | X ( z ) | ∈ , 3 , 1 , ( 1 , 3 ) , ( 3 , 9 ) , ( 9 , ∞ ] . 3 2 πτ πτ Ω 0 π 3 π 2 π − π − πτ − 2 πτ For z ∈ Ω , | X ( z ) | < 1 ⇒ Q ( X ( z ) , 0 ) and Q ( 0 , X ( z )) are well defined. Near Re ( z ) = 0, we have z ∈ Ω and z + πτ ∈ Ω . Counting lattice walks by winding angle Andrew Elvey Price

S OLUTION TO K REWERAS WALKS BY WINDING NUMBER Equation to solve: (near Re ( z ) = 0) R ( X ( z ) , X ( z + πτ )) = 0 where X ( z ) = e − 4 πτ i 3 ϑ ( z , 3 τ ) ϑ ( z − πτ, 3 τ ) ϑ ( z + πτ, 3 τ ) ϑ ( z − 2 πτ, 3 τ ) . R ( x , y ) = 1 − t xQ ( 0 , y ) − t yQ ( x , 0 ) + e i α tQ ( 0 , x ) + e − i α tyQ ( y , 0 ) . Counting lattice walks by winding angle Andrew Elvey Price

S OLUTION TO K REWERAS WALKS BY WINDING NUMBER Equation to solve: (near Re ( z ) = 0) t t 1 = X ( z ) Q ( 0 , X ( z + πτ )) + X ( z + πτ ) Q ( X ( z ) , 0 ) − e i α tQ ( 0 , X ( z )) − e − i α tX ( z + πτ ) Q ( X ( z + πτ ) , 0 ) , where X ( z ) = e − 4 πτ i 3 ϑ ( z , 3 τ ) ϑ ( z − πτ, 3 τ ) ϑ ( z + πτ, 3 τ ) ϑ ( z − 2 πτ, 3 τ ) . Counting lattice walks by winding angle Andrew Elvey Price

S OLUTION TO K REWERAS WALKS BY WINDING NUMBER Equation to solve: (near Re ( z ) = 0) t t 1 = X ( z ) Q ( 0 , X ( z + πτ )) + X ( z + πτ ) Q ( X ( z ) , 0 ) − e i α tQ ( 0 , X ( z )) − e − i α tX ( z + πτ ) Q ( X ( z + πτ ) , 0 ) , where X ( z ) = e − 4 πτ i 3 ϑ ( z , 3 τ ) ϑ ( z − πτ, 3 τ ) ϑ ( z + πτ, 3 τ ) ϑ ( z − 2 πτ, 3 τ ) . For z near 0, define t X ( z + πτ ) Q ( X ( z ) , 0 ) − e i α tQ ( 0 , X ( z )) . L ( z ) = Both L ( z ) and L ( z + πτ ) converge. Counting lattice walks by winding angle Andrew Elvey Price

S OLUTION TO K REWERAS WALKS BY WINDING NUMBER Equation to solve: (near Re ( z ) = 0) t t 1 = X ( z ) Q ( 0 , X ( z + πτ ))+ X ( z + πτ ) Q ( X ( z ) , 0 ) − e i α tQ ( 0 , X ( z )) − e − i α tX ( z + πτ ) Q ( X ( z + πτ ) , 0 ) , where X ( z ) = e − 4 πτ i 3 ϑ ( z , 3 τ ) ϑ ( z − πτ, 3 τ ) ϑ ( z + πτ, 3 τ ) ϑ ( z − 2 πτ, 3 τ ) . For z near 0, define t X ( z + πτ ) Q ( X ( z ) , 0 ) − e i α tQ ( 0 , X ( z )) . L ( z ) = Both L ( z ) and L ( z + πτ ) converge. Counting lattice walks by winding angle Andrew Elvey Price

S OLUTION TO K REWERAS WALKS BY WINDING NUMBER Equation to solve: (near Re ( z ) = 0) t 1 = X ( z ) Q ( 0 , X ( z + πτ )) + L ( z ) − e − i α tX ( z + πτ ) Q ( X ( z + πτ ) , 0 ) , where X ( z ) = e − 4 πτ i 3 ϑ ( z , 3 τ ) ϑ ( z − πτ, 3 τ ) ϑ ( z + πτ, 3 τ ) ϑ ( z − 2 πτ, 3 τ ) . For z near 0, define t X ( z + πτ ) Q ( X ( z ) , 0 ) − e i α tQ ( 0 , X ( z )) . L ( z ) = Both L ( z ) and L ( z + πτ ) converge. Counting lattice walks by winding angle Andrew Elvey Price

S OLUTION TO K REWERAS WALKS BY WINDING NUMBER Equation to solve: (near Re ( z ) = 0) t 1 = X ( z ) Q ( 0 , X ( z + πτ )) + L ( z ) − e − i α tX ( z + πτ ) Q ( X ( z + πτ ) , 0 ) , where X ( z ) = e − 4 πτ i 3 ϑ ( z , 3 τ ) ϑ ( z − πτ, 3 τ ) ϑ ( z + πτ, 3 τ ) ϑ ( z − 2 πτ, 3 τ ) . For z near 0, define t X ( z + πτ ) Q ( X ( z ) , 0 ) − e i α tQ ( 0 , X ( z )) . L ( z ) = Both L ( z ) and L ( z + πτ ) converge. Counting lattice walks by winding angle Andrew Elvey Price

S OLUTION TO K REWERAS WALKS BY WINDING NUMBER Equation to solve: (near Re ( z ) = 0) t 1 = X ( z ) Q ( 0 , X ( z + πτ )) + L ( z ) − e − i α tX ( z + πτ ) Q ( X ( z + πτ ) , 0 ) , where X ( z ) = e − 4 πτ i 3 ϑ ( z , 3 τ ) ϑ ( z − πτ, 3 τ ) ϑ ( z + πτ, 3 τ ) ϑ ( z − 2 πτ, 3 τ ) . For z near 0, define t X ( z + πτ ) Q ( X ( z ) , 0 ) − e i α tQ ( 0 , X ( z )) . L ( z ) = Both L ( z ) and L ( z + πτ ) converge. Counting lattice walks by winding angle Andrew Elvey Price

S OLUTION TO K REWERAS WALKS BY WINDING NUMBER Equation to solve: (near Re ( z ) = 0) t 1 = X ( z ) Q ( 0 , X ( z + πτ )) + L ( z ) e − i α t − X ( z ) X ( z + 2 πτ ) Q ( X ( z + πτ ) , 0 ) , where X ( z ) = e − 4 πτ i 3 ϑ ( z , 3 τ ) ϑ ( z − πτ, 3 τ ) ϑ ( z + πτ, 3 τ ) ϑ ( z − 2 πτ, 3 τ ) . For z near 0, define t X ( z + πτ ) Q ( X ( z ) , 0 ) − e i α tQ ( 0 , X ( z )) . L ( z ) = Both L ( z ) and L ( z + πτ ) converge. Counting lattice walks by winding angle Andrew Elvey Price

S OLUTION TO K REWERAS WALKS BY WINDING NUMBER Equation to solve: (near Re ( z ) = 0) t 1 = X ( z ) Q ( 0 , X ( z + πτ )) + L ( z ) e − i α t − X ( z ) X ( z + 2 πτ ) Q ( X ( z + πτ ) , 0 ) , where X ( z ) = e − 4 πτ i 3 ϑ ( z , 3 τ ) ϑ ( z − πτ, 3 τ ) ϑ ( z + πτ, 3 τ ) ϑ ( z − 2 πτ, 3 τ ) . For z near 0, define t X ( z + πτ ) Q ( X ( z ) , 0 ) − e i α tQ ( 0 , X ( z )) . L ( z ) = Both L ( z ) and L ( z + πτ ) converge. Counting lattice walks by winding angle Andrew Elvey Price

S OLUTION TO K REWERAS WALKS BY WINDING NUMBER Equation to solve: (near Re ( z ) = 0) 1 = − e − i α X ( z ) L ( z + πτ ) + L ( z ) . where X ( z ) = e − 4 πτ i 3 ϑ ( z , 3 τ ) ϑ ( z − πτ, 3 τ ) ϑ ( z + πτ, 3 τ ) ϑ ( z − 2 πτ, 3 τ ) . For z near 0, define t X ( z + πτ ) Q ( X ( z ) , 0 ) − e i α tQ ( 0 , X ( z )) . L ( z ) = Both L ( z ) and L ( z + πτ ) converge. Counting lattice walks by winding angle Andrew Elvey Price

S OLUTION TO K REWERAS WALKS BY WINDING NUMBER Equation to solve: (near Re ( z ) = 0) 1 = − e − i α X ( z ) L ( z + πτ ) + L ( z ) . where X ( z ) = e − 4 πτ i 3 ϑ ( z , 3 τ ) ϑ ( z − πτ, 3 τ ) ϑ ( z + πτ, 3 τ ) ϑ ( z − 2 πτ, 3 τ ) . For z near 0, define t X ( z + πτ ) Q ( X ( z ) , 0 ) − e i α tQ ( 0 , X ( z )) . L ( z ) = Both L ( z ) and L ( z + πτ ) converge. Counting lattice walks by winding angle Andrew Elvey Price

S OLUTION TO K REWERAS WALKS BY WINDING NUMBER Equation to solve: (near Re ( z ) = 0) 1 = − e − i α X ( z ) L ( z + πτ ) + L ( z ) . where X ( z ) = e − 4 πτ i 3 ϑ ( z , 3 τ ) ϑ ( z − πτ, 3 τ ) ϑ ( z + πτ, 3 τ ) ϑ ( z − 2 πτ, 3 τ ) . Counting lattice walks by winding angle Andrew Elvey Price

S OLUTION TO K REWERAS WALKS BY WINDING NUMBER Equation to solve: (near Re ( z ) = 0) 1 = − e − i α X ( z ) L ( z + πτ ) + L ( z ) . where X ( z ) = e − 4 πτ i 3 ϑ ( z , 3 τ ) ϑ ( z − πτ, 3 τ ) ϑ ( z + πτ, 3 τ ) ϑ ( z − 2 πτ, 3 τ ) . We can solve this exactly: e 3 i α 1 + e − i α � � X ( z ) + e − 2 i α X ( z − πτ ) L ( z ) = − 1 − e 3 i α e i α + 5 i πτ ϑ ( z − 2 πτ, 3 τ ) ϑ ( z − α 2 + 2 πτ 3 ϑ ( πτ, 3 τ ) ϑ ′ ( 0 , τ ) 3 , τ ) − ( 1 − e 3 i α ) ϑ ( α 2 − 2 πτ 3 , τ ) ϑ ′ ( 0 , 3 τ ) ϑ ( z , τ ) ϑ ( z , 3 τ ) Counting lattice walks by winding angle Andrew Elvey Price

S OLUTION TO K REWERAS WALKS BY WINDING NUMBER Equation to solve: (near Re ( z ) = 0) 1 = − e − i α X ( z ) L ( z + πτ ) + L ( z ) . where X ( z ) = e − 4 πτ i 3 ϑ ( z , 3 τ ) ϑ ( z − πτ, 3 τ ) ϑ ( z + πτ, 3 τ ) ϑ ( z − 2 πτ, 3 τ ) . We can solve this exactly: e 3 i α 1 + e − i α � � X ( z ) + e − 2 i α X ( z − πτ ) L ( z ) = − 1 − e 3 i α e i α + 5 i πτ ϑ ( z − 2 πτ, 3 τ ) ϑ ( z − α 2 + 2 πτ 3 ϑ ( πτ, 3 τ ) ϑ ′ ( 0 , τ ) 3 , τ ) − ( 1 − e 3 i α ) ϑ ( α 2 − 2 πτ 3 , τ ) ϑ ′ ( 0 , 3 τ ) ϑ ( z , τ ) ϑ ( z , 3 τ ) We can extract E ( t , e i α ) = Q ( 0 , 0 ) ... Counting lattice walks by winding angle Andrew Elvey Price

K REWERAS WALKS BY WINDING NUMBER : S OLUTION Recall: τ is determined by ϑ ′ ( 0 , 3 τ ) t = e − πτ i 4 i ϑ ( πτ, 3 τ ) + 6 ϑ ′ ( πτ, 3 τ ) . 3 The gf E ( t , e i α ) = Q ( 0 , 0 ) ≡ Q ( t , α, 0 , 0 ) is given by: � � 2 − 2 πτ e i α 3 ϑ ( πτ, 3 τ ) ϑ ′ ( α 3 , τ ) ϑ ′ ( 2 πτ, 3 τ ) 4 πτ i πτ i E ( t , e i α ) = e i α − e − e . 3 t ( 1 − e 3 i α ) ϑ ′ ( 0 , 3 τ ) 2 − 2 πτ ϑ ′ ( 0 , 3 τ ) ϑ ( α 3 , τ ) Counting lattice walks by winding angle Andrew Elvey Price

K REWERAS WALKS BY WINDING NUMBER : S OLUTION Recall: τ is determined by ϑ ′ ( 0 , 3 τ ) t = e − πτ i 4 i ϑ ( πτ, 3 τ ) + 6 ϑ ′ ( πτ, 3 τ ) . 3 The gf E ( t , e i α ) = Q ( 0 , 0 ) ≡ Q ( t , α, 0 , 0 ) is given by: � � 2 − 2 πτ e i α 3 ϑ ( πτ, 3 τ ) ϑ ′ ( α 3 , τ ) ϑ ′ ( 2 πτ, 3 τ ) 4 πτ i πτ i E ( t , e i α ) = e i α − e − e . 3 t ( 1 − e 3 i α ) ϑ ′ ( 0 , 3 τ ) 2 − 2 πτ ϑ ′ ( 0 , 3 τ ) ϑ ( α 3 , τ ) Equivalently: Let q ( t ) ≡ q = t 3 + 15 t 6 + 279 t 9 + · · · satisfy T 1 ( 1 , q 3 ) t = q 1 / 3 4 T 0 ( q , q 3 ) + 6 T 1 ( q , q 3 ) . The gf for cell-centred Kreweras-lattice almost-excursions is: � � T 1 ( 1 , q 3 ) − q − 1 / 3 T 0 ( q , q 3 ) T 1 ( sq − 2 / 3 , q ) s − q − 1 / 3 T 1 ( q 2 , q 3 ) s E ( t , s ) = . ( 1 − s 3 ) t T 1 ( 1 , q 3 ) T 0 ( sq − 2 / 3 , q ) Counting lattice walks by winding angle Andrew Elvey Price

Part 3: Walks in cones Counting lattice walks by winding angle Andrew Elvey Price

W ALKS IN CONES WITH SMALL STEPS Quarter plane walks: Completely classified into rational, algebraic, D-finite, D-algebraic cases. [Mishna, Rechnitzer 09], [Bousquet-Mélou, Mishna 10], [Bostan, Kauers 10], [Fayolle, Raschel 10], [Kurkova, Raschel 12], [Melczer, Mishna 13], [Bostan, Raschel, Salvy 14], [Bernardi, Bousquet-Mélou, Raschel 17], [Dreyfus, Hardouin, Roques, Singer 18] Half plane walks: Easy Three quarter plane walks: Active area of research (Previously) solved in 6-12 of the 74 non-trivial cases [Bousquet-Mélou 16], [Raschel-Trotignon 19], [Budd 20], [Bousquet-Mélou, Wallner 20+] Walks on the slit plane C \ R < 0 : solved in all cases [Bousquet-Mélou, 01], [Bousquet-Mélou, Schaeffer, 02], [Rubey 05] Counting lattice walks by winding angle Andrew Elvey Price

W ALKS IN THE 3/4- PLANE : SOLVED CASES D-finite Not D-finite This work [B-M, W 20+] [Budd 20] [R,T 19] [B-M 16] [D,T 20] [Bousquet-Mélou 16],[Raschel, Trotignon 19], [Budd 20], [Bousquet-Mélou, Wallner 20+] Counting lattice walks by winding angle Andrew Elvey Price

W ALKS IN THE 5/4- PLANE : SOLVED CASES D-finite This work [Budd 20] [Budd 20] Counting lattice walks by winding angle Andrew Elvey Price

W ALKS IN THE 6/4- PLANE : SOLVED CASES D-finite This work [Budd 20] [Budd 20] Counting lattice walks by winding angle Andrew Elvey Price

W ALKS IN THE 7/4- PLANE : SOLVED CASES D-finite This work [Budd 20] [Budd 20] Counting lattice walks by winding angle Andrew Elvey Price

C OUNTING K REWERAS WALKS IN A CONE A B R B In the upper half plane: Use reflection principle #( Walks from A to B above R ) = #( Walks from A to B ) − #( Walks from A to B through R ) = #( Walks from A to B ) − #( Walks from A to B ) Counting lattice walks by winding angle Andrew Elvey Price

C OUNTING K REWERAS EXCURSIONS IN 5 / 6- PLANE ✛✻ ❅ ❘ -excursions avoiding a quadrant. New result: ≡ Counting lattice walks by winding angle Andrew Elvey Price

C OUNTING K REWERAS EXCURSIONS IN 5 / 6- PLANE ✛✻ ❅ ❘ -excursions avoiding a quadrant. New result: Equivalently: Walks avoiding the blue and green lines Counting lattice walks by winding angle Andrew Elvey Price

C OUNTING K REWERAS EXCURSIONS IN 5 / 6- PLANE ✛✻ ❅ ❘ -excursions avoiding a quadrant. New result: Equivalently: Walks avoiding the blue and green lines Counting lattice walks by winding angle Andrew Elvey Price

C OUNTING K REWERAS EXCURSIONS IN 5 / 6- PLANE ✛✻ ❅ ❘ -excursions avoiding a quadrant. New result: Equivalently: Walks avoiding the blue and green lines Reflection principle: For walks passing through at least one such line: reflect walk after first intersection. Winding angle α → − 4 π 3 − α or 2 π − α . Counting lattice walks by winding angle Andrew Elvey Price

C OUNTING K REWERAS EXCURSIONS IN 5 / 6- PLANE ✛✻ ❅ ❘ -excursions avoiding a quadrant. New result: Equivalently: Walks avoiding the blue and green lines Reflection principle: For walks passing through at least one such line: reflect walk after first intersection. Winding angle α → − 4 π 3 − α or 2 π − α . Counting lattice walks by winding angle Andrew Elvey Price

C OUNTING K REWERAS EXCURSIONS IN 5 / 6- PLANE ✛✻ ❅ ❘ -excursions avoiding a quadrant. New result: Equivalently: Walks avoiding the blue and green lines Reflection principle: For walks passing through at least one such line: reflect walk after first intersection. Winding angle α → − 4 π 3 − α or 2 π − α . Winding angle 10 π k → − 4 π 3 + 10 π j 3 . 3 #( Walks → avoiding lines ) �� � �� � [ s 5 k ]˜ [ s 5 k − 2 ]˜ = E ( t , s ) − E ( t , s ) k ∈ Z k ∈ Z 4 = 1 � � � � 4 π ij 2 π i � ˜ 1 − e t , e E 5 5 5 j = 1 Counting lattice walks by winding angle Andrew Elvey Price

C OUNTING K REWERAS EXCURSIONS IN 5 / 6- PLANE ✛✻ ❅ ❘ -excursions avoiding a quadrant. New result: Equivalently: Walks avoiding the blue and green lines Reflection principle: For walks passing through at least one such line: reflect walk after first intersection. Winding angle α → − 4 π 3 − α or 2 π − α . Winding angle 10 π k → − 4 π 3 + 10 π j 3 . 3 #( Walks → avoiding lines ) �� � �� � [ s 5 k ]˜ [ s 5 k − 2 ]˜ = E ( t , s ) − E ( t , s ) k ∈ Z k ∈ Z 4 = 1 � � � � 4 π ij 2 π i � ˜ 1 − e t , e E 5 5 5 j = 1 Counting lattice walks by winding angle Andrew Elvey Price

C OUNTING K REWERAS EXCURSIONS IN k / 6- PLANE More generally: Let C k , r ( t ) count whole-plane Kreweras excursions... Starting adjacent to the origin, Avoiding the origin, Having winding angle 0, � � 3 , ( k − r ) π − r π Having intermediate winding angles restricted to 3 I.e., Kreweras excursions in the k / 6-plane Counting lattice walks by winding angle Andrew Elvey Price

C OUNTING K REWERAS EXCURSIONS IN k / 6- PLANE More generally: Let C k , r ( t ) count whole-plane Kreweras excursions... Starting adjacent to the origin, Avoiding the origin, Having winding angle 0, � � 3 , ( k − r ) π − r π Having intermediate winding angles restricted to 3 I.e., Kreweras excursions in the k / 6-plane Previous slide: 4 C 5 , 2 ( t ) = 1 � � � � 4 π ij 2 π i � ˜ 1 − e t , e . E 5 5 5 j = 1 More generally: k − 1 C k , r ( t ) = 1 � 2 π ijr � � 2 π ij � � ˜ 1 − e E t , e . k k k j = 1 Counting lattice walks by winding angle Andrew Elvey Price

Part 4: Analysis of solutions Counting lattice walks by winding angle Andrew Elvey Price

A NALYSIS OF SOLUTION From the exact solution we extract: Asymptotic distribution ([Bélisle, 1989]): For random excursions of length n , winding angle has asymptotic density c log( n ) 4 ( x − 1 ) e x + ( x + 1 ) e − x . ( e x − e − x ) 2 Asymptotics ([Denisov, Wachtel, 2015]): Let c n count Kreweras-lattice excursions in a cone of angle α ∈ π 3 N . k sin 2 � π 2 · 3 5 − 6 � � n − 1 − 3 k k 3 n . c n ∼ − � 2 π − 3 π k 2 � �� � 1 + 2 cos Γ k k Conditions for algebraicity: Let C α ( t ) count Kreweras-lattice excursions in a cone of angle α ∈ π 3 N . This satisfies a non-trivial polynomial equation P ( C α ( t ) , t ) = 0 if and only if α / ∈ π Z . (uses modular forms as in [Zagier, 08] and [E.P., Zinn-Justin, 20]) Counting lattice walks by winding angle Andrew Elvey Price

A NALYSIS OF SOLUTION : A SYMPTOTICS Fix α . τ = − 1 q = e 2 π i ˆ τ , the dominant singularity t = 1 / 3 of Writing ˆ 3 τ and ˆ ˜ E ( t , e i α ) corresponds to ˆ q = 0. Counting lattice walks by winding angle Andrew Elvey Price

A NALYSIS OF SOLUTION : A SYMPTOTICS Fix α . τ = − 1 q = e 2 π i ˆ τ , the dominant singularity t = 1 / 3 of Writing ˆ 3 τ and ˆ ˜ E ( t , e i α ) corresponds to ˆ q = 0. Series in ˆ q : t = 1 q 2 + O (ˆ q 3 ) 3 − 3 ˆ q + 18 ˆ 27 α e i α � � t ˜ 3 α 3 α E ( t , e i α ) = a 0 + a 1 ˆ 2 π + o q − 2 π ( 1 + e i α + e 2 i α )ˆ q ˆ q , 2 π → ˜ E ( t , e i α ) as a series in ( 1 − 3 t ) , Counting lattice walks by winding angle Andrew Elvey Price

A NALYSIS OF SOLUTION : A SYMPTOTICS Fix α . τ = − 1 q = e 2 π i ˆ τ , the dominant singularity t = 1 / 3 of Writing ˆ 3 τ and ˆ ˜ E ( t , e i α ) corresponds to ˆ q = 0. Series in ˆ q : t = 1 q 2 + O (ˆ q 3 ) 3 − 3 ˆ q + 18 ˆ 27 α e i α � � t ˜ 3 α 3 α E ( t , e i α ) = a 0 + a 1 ˆ 2 π + o q − 2 π ( 1 + e i α + e 2 i α )ˆ q ˆ q , 2 π → ˜ E ( t , e i α ) as a series in ( 1 − 3 t ) , → 3 5 − 3 α π e α i α � n − 3 α [ t n ]˜ E ( t , e i α ) ∼ − 2 π − 1 3 n , 2 π ( 1 + e α i + e 2 α i )Γ − 3 α � 2 π k sin 2 � r π 2 · 3 5 − 6 � � n − 1 − 3 [ t n ] C k , r ( t ) ∼ − k k 3 n . � 2 π − 3 π k 2 � �� � 1 + 2 cos Γ k k Counting lattice walks by winding angle Andrew Elvey Price

A NALYSIS OF SOLUTION : A SYMPTOTICS Fix α . τ = − 1 q = e 2 π i ˆ τ , the dominant singularity t = 1 / 3 of Writing ˆ 3 τ and ˆ ˜ E ( t , e i α ) corresponds to ˆ q = 0. Series in ˆ q : t = 1 q 2 + O (ˆ q 3 ) 3 − 3 ˆ q + 18 ˆ 27 α e i α � � t ˜ 3 α 3 α E ( t , e i α ) = a 0 + a 1 ˆ 2 π + o q − 2 π ( 1 + e i α + e 2 i α )ˆ q ˆ q , 2 π → ˜ E ( t , e i α ) as a series in ( 1 − 3 t ) , → 3 5 − 3 α π e α i α � n − 3 α [ t n ]˜ E ( t , e i α ) ∼ − 2 π − 1 3 n , 2 π ( 1 + e α i + e 2 α i )Γ − 3 α � 2 π k sin 2 � r π 2 · 3 5 − 6 � � n − 1 − 3 [ t n ] C k , r ( t ) ∼ − k k 3 n . � 2 π − 3 π k 2 � �� � 1 + 2 cos Γ k k Previously: Terms 3 n and n − 1 − 3 k known [Denisov, Wachtel, 2015]. Counting lattice walks by winding angle Andrew Elvey Price

A NALYSIS OF SOLUTION : A LGEBRAICITY Recall: ϑ ( z , τ ) is differentially algebraic → so are ˜ E ( t , s ) and Q ( t , α, x , y ) . For α ∈ π 3 ( Q \ Z ) we get algebraicity (Ideas from [Zagier, 08] and [E.P., Zinn-Justin, 20+]): Counting lattice walks by winding angle Andrew Elvey Price

A NALYSIS OF SOLUTION : A LGEBRAICITY Recall: ϑ ( z , τ ) is differentially algebraic → so are ˜ E ( t , s ) and Q ( t , α, x , y ) . For α ∈ π 3 ( Q \ Z ) we get algebraicity (Ideas from [Zagier, 08] and [E.P., Zinn-Justin, 20+]): Q ( t , α, X ( z ) , 0 ) and X ( z ) are elliptic functions with the same periods ⇒ Q ( t , α, x , 0 ) is algebraic in x . Counting lattice walks by winding angle Andrew Elvey Price

A NALYSIS OF SOLUTION : A LGEBRAICITY Recall: ϑ ( z , τ ) is differentially algebraic → so are ˜ E ( t , s ) and Q ( t , α, x , y ) . For α ∈ π 3 ( Q \ Z ) we get algebraicity (Ideas from [Zagier, 08] and [E.P., Zinn-Justin, 20+]): Q ( t , α, X ( z ) , 0 ) and X ( z ) are elliptic functions with the same periods ⇒ Q ( t , α, x , 0 ) is algebraic in x . E ( t ( τ ) , e i α ) and t ( τ ) are modular functions of τ ⇒ E ( t , e i α ) is algebraic in t . Same for ˜ E ( t ( τ ) , e i α ) . Counting lattice walks by winding angle Andrew Elvey Price

A NALYSIS OF SOLUTION : A LGEBRAICITY Recall: ϑ ( z , τ ) is differentially algebraic → so are ˜ E ( t , s ) and Q ( t , α, x , y ) . For α ∈ π 3 ( Q \ Z ) we get algebraicity (Ideas from [Zagier, 08] and [E.P., Zinn-Justin, 20+]): Q ( t , α, X ( z ) , 0 ) and X ( z ) are elliptic functions with the same periods ⇒ Q ( t , α, x , 0 ) is algebraic in x . E ( t ( τ ) , e i α ) and t ( τ ) are modular functions of τ ⇒ E ( t , e i α ) is algebraic in t . Same for ˜ E ( t ( τ ) , e i α ) . Combining these ideas: Q ( t , α, x , y ) is algebraic in t , x and y . Counting lattice walks by winding angle Andrew Elvey Price

A NALYSIS OF SOLUTION : A LGEBRAICITY Recall: ϑ ( z , τ ) is differentially algebraic → so are ˜ E ( t , s ) and Q ( t , α, x , y ) . For α ∈ π 3 ( Q \ Z ) we get algebraicity (Ideas from [Zagier, 08] and [E.P., Zinn-Justin, 20+]): Q ( t , α, X ( z ) , 0 ) and X ( z ) are elliptic functions with the same periods ⇒ Q ( t , α, x , 0 ) is algebraic in x . E ( t ( τ ) , e i α ) and t ( τ ) are modular functions of τ ⇒ E ( t , e i α ) is algebraic in t . Same for ˜ E ( t ( τ ) , e i α ) . Combining these ideas: Q ( t , α, x , y ) is algebraic in t , x and y . Recall: The gf for excursions in the k / 6-plane is k − 1 C k , r ( t ) = 1 � 2 π ijr � � 2 π ij � � ˜ 1 − e E t , e . k k k j = 1 Counting lattice walks by winding angle Andrew Elvey Price

A NALYSIS OF SOLUTION : A LGEBRAICITY Recall: ϑ ( z , τ ) is differentially algebraic → so are ˜ E ( t , s ) and Q ( t , α, x , y ) . For α ∈ π 3 ( Q \ Z ) we get algebraicity (Ideas from [Zagier, 08] and [E.P., Zinn-Justin, 20+]): Q ( t , α, X ( z ) , 0 ) and X ( z ) are elliptic functions with the same periods ⇒ Q ( t , α, x , 0 ) is algebraic in x . E ( t ( τ ) , e i α ) and t ( τ ) are modular functions of τ ⇒ E ( t , e i α ) is algebraic in t . Same for ˜ E ( t ( τ ) , e i α ) . Combining these ideas: Q ( t , α, x , y ) is algebraic in t , x and y . Recall: The gf for excursions in the k / 6-plane is k − 1 C k , r ( t ) = 1 � 2 π ijr � � 2 π ij � � ˜ 1 − e E t , e . k k k j = 1 Algebraic iff 3 ∤ k . (always D-finite). Counting lattice walks by winding angle Andrew Elvey Price

Part 5: Other lattices Kreweras lattice Triangular Lattice Square Lattice King Lattice Counting lattice walks by winding angle Andrew Elvey Price

CELL - CENTRED LATTICES Important property: Decomposable into congruent sectors Kreweras lattice Triangular Lattice Square Lattice King Lattice Counting lattice walks by winding angle Andrew Elvey Price

CELL - CENTRED LATTICES Important property: Decomposable into congruent sectors Kreweras lattice Triangular Lattice Square Lattice King Lattice Counting lattice walks by winding angle Andrew Elvey Price

VERTEX - CENTRED LATTICES Decompose into rotationally congruent sectors Kreweras lattice Triangular Lattice Square Lattice King Lattice Counting lattice walks by winding angle Andrew Elvey Price

VERTEX - CENTRED LATTICES Decompose into rotationally congruent sectors Kreweras lattice Triangular Lattice Square Lattice King Lattice Counting lattice walks by winding angle Andrew Elvey Price

R ECALL : K REWERAS ALMOST - EXCURSIONS ∞ ( − 1 ) n ( 2 n + 1 ) k q n ( n + 1 ) / 2 ( u n + 1 − ( − 1 ) k u − n ) � Define T k ( u , q ) = n = 0 = ( u ± 1 ) − 3 k q ( u 2 ± u − 1 ) + 5 k q 3 ( u 3 ± u − 2 ) + O ( q 6 ) . Let q ( t ) ≡ q = t 3 + 15 t 6 + 279 t 9 + · · · satisfy T 1 ( 1 , q 3 ) t = q 1 / 3 4 T 0 ( q , q 3 ) + 6 T 1 ( q , q 3 ) . The gf for cell-centred Kreweras-lattice almost-excursions is: � � s − q − 1 / 3 T 1 ( q 2 , q 3 ) T 1 ( 1 , q 3 ) − q − 1 / 3 T 0 ( q , q 3 ) T 1 ( sq − 2 / 3 , q ) s E ( t , s ) = . ( 1 − s 3 ) t T 1 ( 1 , q 3 ) T 0 ( sq − 2 / 3 , q ) The gf for vertex-centred Kreweras-lattice almost-excursions is: � T 1 ( q , q 3 ) 2 E ( t , s ) = s ( 1 − s ) q − 2 T 0 ( q , q 3 ) 2 T 0 ( q , q 3 ) 2 − T 2 ( q , q 3 ) 6 T 1 ( 1 , q ) + T 3 ( 1 , q 3 ) T 0 ( q , q 3 ) − T 2 ( s , q ) 2 T 0 ( s , q ) + T 3 ( 1 , q ) � 3 ˜ . t ( 1 − s 3 ) T 1 ( 1 , q 3 ) 2 3 T 1 ( 1 , q 3 ) Counting lattice walks by winding angle Andrew Elvey Price

S QUARE LATTICE ALMOST - EXCURSIONS ∞ ( − 1 ) n ( 2 n + 1 ) k q n ( n + 1 ) / 2 ( u n + 1 − ( − 1 ) k u − n ) � Define T k ( u , q ) = n = 0 = ( u ± 1 ) − 3 k q ( u 2 ± u − 1 ) + 5 k q 3 ( u 3 ± u − 2 ) + O ( q 6 ) . Let q ( t ) ≡ q = t + 4 t 3 + 34 t 5 + 360 t 7 + · · · satisfy qT 0 ( q 2 , q 8 ) T 1 ( 1 , q 8 ) t = 2 T 0 ( q 4 , q 8 )( T 0 ( q 2 , q 8 ) + 2 T 1 ( q 2 , q 8 )) . The gf for cell-centred Square-lattice almost-excursions is: s 2 s − s − 1 + T 0 ( q 4 , q 8 ) qT 1 ( 1 , q 8 ) − T 0 ( q 4 , q 8 ) T 1 ( s − 1 q , q 2 ) � � . qT 1 ( 1 , q 8 ) T 0 ( s − 1 q , q 2 ) ( 1 − s 4 ) t The gf for vertex-centred Square-lattice almost-excursions is: sT 0 ( q 4 , q 8 ) 1 + 2 T 1 ( q 2 , q 8 ) T 0 ( q 2 , q 8 ) + ( 1 − s ) T 1 ( s − 1 , q 2 ) � � . ( 1 + s ) T 0 ( s − 1 , q 2 ) qt ( 1 + s 2 ) T 1 ( 1 , q 8 ) Counting lattice walks by winding angle Andrew Elvey Price

Part 6: Final comments Counting lattice walks by winding angle Andrew Elvey Price

J ACOBI THETA FUNCTION / W EIERSTRASS FUNCTION PARAMETERISATION COMBINATORIAL FUNCTIONAL EQUATION SOLUTION METHOD Counting lattice walks by winding angle Andrew Elvey Price