Counting quadrant walks via Tutte s invariant method

Counting quadrant walks via Tutte s invariant method Olivier Bernardi - Brandeis University Mireille Bousquet-Mélou - CNRS, Université de Bordeaux Kilian Raschel - CNRS, Université de Tours Vancouver, FPS 2016

Quadrant walks W S = set of lattice walks on N 2 starting at (0, 0) with steps in S, for some S { 1, 0, 1} 2. Example: S = w W S

Quadrant walks W S = set of lattice walks on N 2 starting at (0, 0) with steps in S, for some S { 1, 0, 1} 2. Example: S = w W S Generating function: Q(x, y) = x i(w) y j(w) t w, w W S where (i(w), j(w)) is the ending point and w is the number of steps.

Equation for Q(x, y) = w W S x i(w) y j(w) t w. Q(x, y) = 1 + ( t (i,j) S x i y j )Q(x, y)...

Equation for Q(x, y) = w W S x i(w) y j(w) t w. Q(x, y) = 1 + ( t x i y j )Q(x, y)... (i,j) S Functional equation: where ( K(x, y) = xy t K(x, y)q(x, y) = R(x) + S(y) xy, (i,j) S ) x i y j 1, Kernel R(x)=K(x, 0)Q(x, 0) K(0, 0)Q(0, 0), and S(y)=K(0, y)q(0, y).

Goals: Find a better equation for Q(x, y) (or even just for Q(0, 0)).

Goals: Find a better equation for Q(x, y) (or even just for Q(0, 0)). Determine the nature of Q(x, y): Algebraic? Pol(Q(x, y), x, y, t) = 0. D-finite? k p k(x, y, t)( t )k Q(x, y)=0 + same for x, y. D-Algebraic? Pol(Q(x, y), tq(x, y),..., x, y, t) = 0.

Goals: Find a better equation for Q(x, y) (or even just for Q(0, 0)). Determine the nature of Q(x, y): From the previous episodes... Algebraic approach: [G86, GZ92, B-M02, B-MMi10] Computer approach: [KaZ08, KaKoZ09, BKa10, KaY15] Analytic approach: [FIMa99, BKuRa13, R12, KuR12] Asymptotic approach: [MiRe09, BRaS12, DW15] Starring: Bostan, Bousquet-Mélou, Denisov, Fayolle, Gessel, Iasnogorodski, Kauers, Koutschan, Kurkova, Malyshev, Mishna, Raschel, Rechnitzer, Salvy, Watchel, Yatchak, Zeilberger...

Goals: Find a better equation for Q(x, y) (or even just for Q(0, 0)). Determine the nature of Q(x, y): From the previous episodes... There are 79 non-isomorphic, non-trivial choices of S. Algebraic D-finite not algebraic not D-finite 4 19 56

Goals: Find a better equation for Q(x, y) (or even just for Q(0, 0)). Determine the nature of Q(x, y): From the previous episodes... There are 79 non-isomorphic, non-trivial choices of S. Algebraic D-finite not algebraic not D-finite 4 19 56 Q(x, y) is D-finite group Γ is finite. ( ) ( c(y) Γ =< φ, ψ > where φ(x, y)= ã(y)x, y c(x) and ψ(x, y)= x, a(x)y where K(x, y) = ã(y)x 2 + b(y)x + c(y) = a(x)y 2 + b(x)y + c(x). )

Goals: Find a better equation for Q(x, y) (or even just for Q(0, 0)). Determine the nature of Q(x, y): From the previous episodes... There are 79 non-isomorphic, non-trivial choices of S. Algebraic D-finite not algebraic not D-finite 4 19 56 Q(x, y) is D-finite group Γ is finite. Q(x, y) is algebraic γ Γ( 1) γ γ(xy) = 0.

Goals: Find a better equation for Q(x, y) (or even just for Q(0, 0)). Determine the nature of Q(x, y): From the previous episodes... There are 79 non-isomorphic, non-trivial choices of S. Algebraic D-finite not algebraic not D-finite 4 19 56 Q(x, y) is D-finite group Γ is finite. Q(x, y) is algebraic γ Γ( 1) γ γ(xy) = 0. Expressions of Q(x, y) in terms of Weierstrass elliptic function.

Our contributions Simpler, uniform, proofs for the 4 algebraic models. S = Gessel Model

Our contributions Simpler, uniform, proofs for the 4 algebraic models. S = Gessel Model Extension to 4 algebraic weighted models. 1 1 2 1 1 1 1 2 1 S = 1 2 1 2 2 1 2 1 1 λ 1 1 1 1 2 1 1 1 New! (conjectured Kauers & Yatchak)

Our contributions Simpler, uniform, proofs for the 4 algebraic models. S = Gessel Model Extension to 4 algebraic weighted models. 1 1 2 1 1 1 1 2 1 S = 1 2 1 2 2 1 2 1 1 λ 1 1 1 1 2 1 1 1 New! (conjectured Kauers & Yatchak) Proof of D-algebraicity + simpler expressions for 9 non-d-finite models.

A source of inspiration William Tutte G(x, y) = x(q 1) + xytg(1, y)g(x, y) x 2 G(x, y) G(1, y) yt + xt x 1 [Tutte: Chromatic Sum Revisited (95)] G(x, y) G(x, 0) y.

Invariant method - Case of Gessel walks

Sketch of the invariant method Functional equation: ( K(x, y)q(x, y) = R(x) + S(y) xy, ( ) where K(x, y) = xy t ) x i y j 1 and S(y)=K(0, y)q(0, y). (i,j) S

Sketch of the invariant method Functional equation: ( K(x, y)q(x, y) = R(x) + S(y) xy, ( ) where K(x, y) = xy t ) x i y j 1 and S(y)=K(0, y)q(0, y). (i,j) S Let Y 1 Y 1 (x) and Y 2 Y 2 (x) be such that K(x, Y i ) = 0.

Sketch of the invariant method Functional equation: ( K(x, y)q(x, y) = R(x) + S(y) xy, ( ) where K(x, y) = xy t ) x i y j 1 and S(y)=K(0, y)q(0, y). (i,j) S Let Y 1 Y 1 (x) and Y 2 Y 2 (x) be such that K(x, Y i ) = 0. { K(x, Y 1 ) = 0 = K(x, Y 2 ), ( ) S(Y 1 ) xy 1 = R(x) = S(Y 2 ) xy 2.

Sketch of the invariant method Functional equation: ( K(x, y)q(x, y) = R(x) + S(y) xy, ( ) where K(x, y) = xy t ) x i y j 1 and S(y)=K(0, y)q(0, y). (i,j) S Let Y 1 Y 1 (x) and Y 2 Y 2 (x) be such that K(x, Y i ) = 0. { K(x, Y 1 ) = 0 = K(x, Y 2 ), ( ) S(Y 1 ) xy 1 = R(x) = S(Y 2 ) xy 2. Def. An invariant is a series I(y) such that I(Y 1 ) = I(Y 2 ).

Sketch of the invariant method Functional equation: ( K(x, y)q(x, y) = R(x) + S(y) xy, ( ) where K(x, y) = xy t ) x i y j 1 and S(y)=K(0, y)q(0, y). (i,j) S Let Y 1 Y 1 (x) and Y 2 Y 2 (x) be such that K(x, Y i ) = 0. { K(x, Y 1 ) = 0 = K(x, Y 2 ), ( ) S(Y 1 ) xy 1 = R(x) = S(Y 2 ) xy 2. Def. An invariant is a series I(y) such that I(Y 1 ) = I(Y 2 ). { I(Y 1 ) = I(Y 2 ), with I(y) rational S(Y 1 ) G(Y 1 ) = S(Y 2 ) G(Y 2 ), with G(y) is rational

Sketch of the invariant method Functional equation: ( K(x, y)q(x, y) = R(x) + S(y) xy, ( ) where K(x, y) = xy t ) x i y j 1 and S(y)=K(0, y)q(0, y). (i,j) S Let Y 1 Y 1 (x) and Y 2 Y 2 (x) be such that K(x, Y i ) = 0. { K(x, Y 1 ) = 0 = K(x, Y 2 ), ( ) S(Y 1 ) xy 1 = R(x) = S(Y 2 ) xy 2. Def. An invariant is a series I(y) such that I(Y 1 ) = I(Y 2 ). { I(Y 1 ) = I(Y 2 ), with I(y) rational S(Y 1 ) G(Y 1 ) = S(Y 2 ) G(Y 2 ), with G(y) is rational Def. A decoupling function is a series G(y) such that x Y 1 x Y 2 = G(Y 1 ) G(Y 2 ).

Sketch of the invariant method Functional equation: ( K(x, y)q(x, y) = R(x) + S(y) xy, ( ) where K(x, y) = xy t ) x i y j 1 and S(y)=K(0, y)q(0, y). (i,j) S Let Y 1 Y 1 (x) and Y 2 Y 2 (x) be such that K(x, Y i ) = 0. { K(x, Y 1 ) = 0 = K(x, Y 2 ), ( ) S(Y 1 ) xy 1 = R(x) = S(Y 2 ) xy 2. Def. An invariant is a series I(y) such that I(Y 1 ) = I(Y 2 ). { I(Y1 ) = I(Y 2 ), J(Y 1 ) = J(Y 2 ), where J(y) = S(y) G(y). Equation between I(y) and J(y) (i.e. equation for S(y)). Invariant Lemma

Example: Invariant method for Gessel walks Functional equation: K(x, y)q(x, y) = R(x) + S(y) xy, ( ) where K(x, y)=tx 2 y 2 +(tx 2 x+t)y+t and S(y)=t(1 + y)q(0, y).

Example: Invariant method for Gessel walks Functional equation: K(x, y)q(x, y) = R(x) + S(y) xy, ( ) where K(x, y)=tx 2 y 2 +(tx 2 x+t)y+t and S(y)=t(1 + y)q(0, y). ( ) { K(x, Y1 ) = 0 = K(x, Y 2 ), S(Y 1 ) xy 1 = S(Y 2 ) xy 2. ( ) Upon setting x = t + t 2 (u + 1 u )

Example: Invariant method for Gessel walks Functional equation: K(x, y)q(x, y) = R(x) + S(y) xy, ( ) where K(x, y)=tx 2 y 2 +(tx 2 x+t)y+t and S(y)=t(1 + y)q(0, y). ( ) { K(x, Y1 ) = 0 = K(x, Y 2 ), S(Y 1 ) xy 1 = S(Y 2 ) xy 2. y I(Y 1 ) = I(Y 2 ), for I(y) = t(1+y) 2 + t(1+y)2. y J(Y 1 ) = J(Y 2 ), for J(y) = S(y) + 1 t(1+y). rational invariant decoupling function

Example: Invariant method for Gessel walks Invariant Lemma: If a series P (y) = n= n 0 n+n 0 j=0 is an invariant, then P (y) is in fact independent of y. a j,n y j t 2n (J(y) J(0))I(y) + a J(y) 3 + b J(y) 2 + c J(y) + d = 0, with J(y), a, b, c, d given in terms of S(y).

Example: Invariant method for Gessel walks Invariant Lemma: If a series P (y) = ( ) ( ) S (y) + 1 t(y+1) S (0) 1 1 t t(y+1)(1+1/y) ( ) + t (y + 1 + 1/y) 3 ( ) 2 +t S (y) + 1 t(y+1) (2 + ts (0)) S (y) + 1 t(y+1) ( ) ts(0)+2 S ( 1)t t 2 1 S (y) + 1 t t(y+1) + 2 S(0)S ( 1)t S(0)t 2 S ( 1)+3 S ( 1) t = 0. t n= n 0 n+n 0 j=0 is an invariant, then P (y) is in fact independent of y. a j,n y j t 2n

Example: Invariant method for Gessel walks Invariant Lemma: If a series P (y) = ( ) ( ) S (y) + 1 t(y+1) S (0) 1 1 t t(y+1)(1+1/y) ( ) + t (y + 1 + 1/y) 3 ( ) 2 +t S (y) + 1 t(y+1) (2 + ts (0)) S (y) + 1 t(y+1) ( ) ts(0)+2 S ( 1)t t 2 1 S (y) + 1 t t(y+1) + 2 S(0)S ( 1)t S(0)t 2 S ( 1)+3 S ( 1) t = 0. t n= n 0 n+n 0 j=0 is an invariant, then P (y) is in fact independent of y. a j,n y j t 2n [Bousquet-Mélou, Jehanne 06] Algebraic equation for S(y) (hence for Q(x, y)).

Summary Invariant method requires: Rational invariant I(y): I(Y 1 ) = I(Y 2 ). Decoupling function G(y): xy 1 xy 2 = G(Y 1 ) G(Y 2 ). Invariant Lemma. It gives: Algebraic equation for S(y).

Algebraic models Rational invariant I(y) t/y 2 1/y ty ty 2 y t/y Decoupling function G(y) 1/y 1/y Invariant Lemma Gessel 1 1 2 1 λ 1 1 1 2 1 2 1 1 1 2 1 1 1 2 1 1 2 2 1 1 1 1 t/y ty 1+2t y+1 y t(1+y) 2 + t(1+y)2 y ( t 2 y + 1+λt y+1 1+λt y+1 t(t y(2t+1) y 3 (3t+1)) y 2 (1+3t)(4t+1) y+1 ) 2 + (3t+1)2 (y+1) 2 t(ty 3 y 2 (2t+1) (3t+1)) y (1+3t)(4t+1) + (3t+1)2 1/y+1 (1/y+1) 2 t(t y(2t+1) y 3 (3t+1)) y 2 (1+3t)(4t+1) y+1 + (3t+1)2 (y+1) 2 1/y 1/t(1 + y) (1 + λt)/t(1 + y) 1/y y (1+3t)/t(1+y) y(1 y+1/t) (1+3t)/ty y + 1/y

Other models? Prop. For the 79 unweighted models, Rational invariant I(y) exists Group Γ is finite Q(x, y) is D-finite.

Other models? Prop. For the 79 unweighted models, Rational invariant I(y) exists Group Γ is finite Q(x, y) is D-finite. Prop. For models such that the group Γ is finite, Decoupling function G(y) exists γ Γ ( 1) γ γ(xy) = 0 Q(x, y) is algebraic. Moreover, the decoupling function can be found as a sum over Γ.

Other models? Prop. For the 79 unweighted models, Rational invariant I(y) exists Group Γ is finite Q(x, y) is D-finite. Prop. For models such that the group Γ is finite, Decoupling function G(y) exists γ Γ ( 1) γ γ(xy) = 0 Q(x, y) is algebraic. Moreover, the decoupling function can be found as a sum over Γ. Prop. Among the 56 unweighted models such that group Γ is infinite, exactly 9 have a decoupling function.

Analytic invariant method

Weak invariants Let Y i (x) = b(x)± (x) 2a(x) be such that K(x, Y i (x))) = 0.

Weak invariants Let Y i (x) = b(x)± (x) 2a(x) be such that K(x, Y i (x))) = 0. Fact [FIM99]: For all t [0, 1/ S ], (x) -1 x 1 x 2 x 3 x 4 or + 1 x

Weak invariants Let Y i (x) = b(x)± (x) 2a(x) be such that K(x, Y i (x))) = 0. Fact [FIM99]: For all t [0, 1/ S ], (x) -1 x 1 x 2 x 3 x 4 or + 1 x Let L = Y 1 ([x 1, x 2 ]) Y 2 ([x 1, x 2 ]). G L G L

Weak invariants Let Y i (x) = b(x)± (x) 2a(x) be such that K(x, Y i (x))) = 0. Fact [FIM99]: For all t [0, 1/ S ], (x) -1 x 1 x 2 x 3 x 4 or + 1 x Let L = Y 1 ([x 1, x 2 ]) Y 2 ([x 1, x 2 ]). Def. Fix t < 1/ S. A weak invariant is a function I(y) such that I(y) is meromorphic in G L and has finite values on L. x [x 1, x 2 ], I(Y 1 (x)) = I(Y 2 (x)).

Analytic invariant method Thm. For all S { 1, 0, 1} 2 there exists a weak invariant W (y) which is moreover injective in G L (conformal map) and D-algebraic.

Analytic invariant method Thm. For all S { 1, 0, 1} 2 there exists a weak invariant W (y) which is moreover injective in G L (conformal map) and D-algebraic. W (y) = ω1,ω 3 ( ω 1 + ω 2 2 + 1 ω 1,ω 2 ( a + b )), y c where a, b, c are explicit constants (algebraic in t), ω 1 = i x2 dx (x), ω 2 = x3 dx (x), ω 3 = x1 dx (x). x 1 x 2 Y (x 1 ) and α,β (z) is Weierstrass elliptic function: α,β (z) = 1 z 2 + (m,n) Z 2 \{(0,0)} 1 (z + mα + nβ) 2 1 (mα + nβ) 2.

Analytic invariant method Thm. For all S { 1, 0, 1} 2 there exists a weak invariant W (y) which is moreover injective in G L (conformal map) and D-algebraic. Thm. Let S { 1, 0, 1} be a non-singular model. If I(y) is a weak invariant and is bounded on G L, then I(y) is independent of y.

Analytic invariant method Thm. For all S { 1, 0, 1} 2 there exists a weak invariant W (y) which is moreover injective in G L (conformal map) and D-algebraic. Thm. Let S { 1, 0, 1} be a non-singular model. If I(y) is a weak invariant and is bounded on G L, then I(y) is independent of y. Applications: For the last algebraic model: an algebraic equation for S(y). (from 2 algebraic equations between I(y), J(y), and W (y)). For the 9 non-algebraic models having a decoupling function: Expression of S(y) in terms of W (y) (+ proof of D-algebraicity).

Open Questions

1. Decoupling functions No decoupling function Finite group Algebraic D-finite transcendental Infinite group Differentially algebraic??? 2. For the 9 non-algebraic models having a decoupling function: What are the D-algebraic equations? 3. Is there a bijective proof for the number of Gessel walks? q n = 16 n (1/2) n(5/6) n (2) n (5/3) n, where (a) n = a(a + 1) (a + n 1).

Thanks.