Abstract
In the 1970s, William Tutte developed a clever algebraic approach, based on certain "invariants", to solve a functional equation that arises in the enumeration of properly colored triangulations. The enumeration of plane lattice walks confined to the first quadrant is governed by similar equations, and has led in the past 20 years to a rich collection of attractive results dealing with the nature (algebraic, D-finite or not) of the associated generating function, depending on the set of allowed steps, taken in \(\{-1, 0,1\}^2\).We first adapt Tutte's approach to prove (or reprove) the algebraicity of all quadrant models known or conjectured to be algebraic. This includes Gessel's famous model, and the first proof ever found for one model with weighted steps. To be applicable, the method requires the existence of two rational functions called invariant and decoupling function respectively. When they exist, algebraicity follows almost automatically.Then, we move to a complex analytic viewpoint that has already proved very powerful, leading in particular to integral expressions for the generating function in the non-D-finite cases, as well as to proofs of non-D-finiteness. We develop in this context a weaker notion of invariant. Now all quadrant models have invariants, and for those that have in addition a decoupling function, we obtain integral-free expressions for the generating function, and a proof that this series is D-algebraic (that is, satisfies polynomial differential equations).Keywords: Lattice walks, enumeration, differentially algebraic series, conformal mappings.Mathematics Subject Classifications: 05A15, 34K06, 39A06, 30C20, 30D05
Highlights
We prove that invariants exist if and only if the group of the model is finite (that is, if and only if Q(x, y; t) is D-finite); this happens for 23 models
The existence of invariants only depends on the step set S, but the existence of decoupling functions is sensitive to the starting point: in Section 7, we describe for which points they exist
By subtracting an appropriate series of the form C1L(y)3 + C2L(y)2 + C3L(y), where C1, C2 and C3 depend on t but not on y, we obtain a Laurent series in t satisfying the assumptions of the lemma: the polynomiality of the coefficients in y holds by construction, and the fact that in each monomial yjtn, the exponent of j is at most half the exponent of n comes from the fact that this holds in S(y), due to the choice of the step set (a walk ending at (0, j) has at least 2j steps)
Summary
Dreyfus, Hardouin, Roques and Singer completed the differential classification of quadrant walks by proving that the remaining 47 infinite group models are neither D-algebraic in x (nor in y, by symmetry) [DHRS18, DHRS20], nor in t [DH19] (see the recent preprint [HS20] on walks with weighted steps). Their proofs rely on Galois theory for difference equations. One key step in his study was to prove that for certain (infinitely many) values of q, the series G(x, y) is algebraic, using a pair of (non-rational) series that he called invariants [Tut95] They are replaced in our approach by (rational) invariants and decoupling functions. The notion of invariants appears as one way to bridge the gap between the algebraic and analytic approaches to quadrant walks
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