For generalized Dyck paths (i.e., directed lattice paths with any nite set of jumps), we analyse their local time at zero (i.e., the number of times the path is touching or crossing the abscissa). As we are in a discrete setting, the event we analyse here is \invisible" to the tools of Brownian motion theory. It is interesting that the key tool for analysing directed lattice paths, which is the kernel method, is not directly applicable here. Therefore, we introduce a variant of this kernel method to get the trivariate generating function (length, nal altitude, local time): this leads to an expression involving symmetric and algebraic functions. We apply this analysis to di erent types of constrained lattice paths (meanders, excursions, bridges, . . . ). Then, we illustrate this approach on \basketball walks" which are walks de ned by the jumps 2; 1; 0; +1; +2. We use singularity analysis to prove that the limit laws for the local time are (depending on the drift and the type of walk) the geometric distribution, the negative binomial distribution, the Rayleigh distribution, or the half-normal distribution (a universal distribution up to now rarely encountered in analytic combinatorics).
This article continues our series of investigations on the enumeration, generation, and asymptotics of directed lattice paths [ABBG18a, ABBG18b, BF02, BW14, BW17, BG06, BKKKKNW17a, Wal16]. Such lattice paths are a fundamental combinatorial structure ubiquitous in computer science (evolution of a stack, bijections with trees, permutations, . . . ), probability theory (linked with random walks or queuing theory), and statistical mechanics (as basic building blocks for more general 2D models), to name a few. unconstrained
(on Z) constrained (on N) ending anywhere meander (M)
bridge (B) excursion (E)
Let us give a de nition of the lattice paths we consider: De nition 1.1 (Jumps and lattice paths). A step set S Z2 is a nite set of vectors f(x1; y1); : : : ; (xm; ym)g. An n-step lattice path or walk is a sequence of vectors (v1; : : : ; vn), such that vj is in S. Geometrically, it may be interpreted as a sequence of points ! = (!0; !1; : : : ; !n) where !i 2 Z2; !0 = (0; 0) (or another starting point) and !i !i 1 = vi for i = 1; : : : ; n. The elements of S are called steps or jumps. The length j!j of a lattice path is its number n of jumps.
We restrict our attention to directed paths which are de ned by the fact that, for each jump (x; y) 2 S, one must have x 0. Lattice paths can have di erent additional constraints shown in Table 1.
Note that it is possible to encode lattice paths in words. Then, the constrained lattice paths we consider can be enumerated by context-free grammars [LY90, MRSV99, Duc00]. One drawback of the grammar approach is that it is not easy to get universal asymptotic results from it, even if it is possible to establish generic results on the associated critical exponents [BD15]. Also, the grammar approach quickly leads to heavy case-by-case computations (resultants of equations of huge degree) as soon as the set of jumps S contains a large jump.
In this article, we show how to proceed for the enumeration and the asymptotics in these harder cases: our techniques are relying on the \kernel method" which (contrary to the context-free grammar approach) o ers access to the generic structure of the nal generating functions and the universality of their asymptotics via singularity analysis [BF02, FS09].
The following convenient notation, a variant of the Omega operator of MacMahon, will be another of our ingredients: fu>0g :=
X un[un] n>0 and fu<0g :=
X un[un] n<0 where [un]g(u) stands for the coe cient of un in a (Laurent) power series g(u) from C((u)) or C[u; 1=u][[z]]. MacMahon introduced this operator on rational functions, in order to get binomial identities or integer partition formulae [Mac15, Section VIII]. In the late 1990s, this operator experienced a strong revival, mostly by work/packages of Andrews, Paule, Riese, and Han [APR01, Han03]. Another nice application of the Omega operator is the proof of D- niteness of some walks in the quarter plane by Bousquet-Melou and Mishna [BM10]. In this article we make use of this operator, not on sums or products of rational functions (like it is the case for integer partitions, or for quarter plane walks), but on the level of functional equations involving algebraic functions.
Generating function for the local time at 0 The number of times the lattice path is exactly at altitude 0 is an easy parameter to catch via combinatorial decompositions (see analysis of the number of returns to zero for excursions, via a decomposition into arches, by Banderier and Flajolet [BF02]). In order to get the local time at 0 (as de ned in Figure 1), it remains to capture a more subtle parameter: the number of steps which are crossing the x-axis (without actually starting or ending at altitude 0). For any family of lattice paths, let P (u) = 1=uc + + ud be the Laurent polynomial encoding the jumps allowed at each step. For example, the \basketball walks" which we considered in [BKKKKNW17a] are walks with jumps in the set f 2; 1; 0; +1; +2g, and we get P (u) := 1=u2 + 1=u + 1 + u + u2. More generally, each jump i may get a weight pi, which gives d P (u) = X piui:
i= c
Feller [Fel68, Fel71], Czaki and Vincze [CV61] and Jain [Jai66], considered P (u) = u=2 + (1=2)u, Wallner [Wal16] considered P (u) = p 1=u + p0 + p1u, and we show here which new method is needed to tackle more general P (u).
As the generating function of the returns to zero, and the corresponding limit laws are known (see [BF02,Wal16]), we can now focus on the number of x-axis crossings.
Theorem 2.1 (Generating function for number of x-axis crossings). Let wn;k;j be the number of walks of length n which end at altitude k, and have j crossings of the x-axis. Then the generating function W (z; u; q) =
X n;j 0;k2Z wn;k;j znukqj = X Wk(z; q)uk k2Z is algebraic and expressible in terms of the roots ui(z) of 1 zP (u) = 0.
Proof (Sketch). A step by step decomposition of the walk gives the following functional equation (where we write Wk for Wk(z; q) for readability, and where q encodes the x-axis crossings):
W (z; u; q) = 1 + zP (u)W (z; u; q)
z + zq
k=1 This equation can be read as \W = empty walk or the walk is at some altitude encoded by uk and multiply by P (u) to do all the jumps, then we remove the ones crossing the x-axis, and re-add them with a marker q." This is conveniently rewritten as
X Wkuk k2Z
! (1 zP (u))
Here, one may think that we could apply the classical kernel method (see e.g. [BF02]): one substitutes u by any root u(z) of 1 zP (u), as this will cancel the left-hand side, and thus doing that for all roots will lead to a system of c + d unknowns, with c + d independent relations. Hence, bingo, solving this system gives a closed-form formula for W (z; u; q)!? Unfortunately, this is not working: indeed, in such an equation, if one substitutes a variable by another expression, one has to take care to stay in the ring of formal power series (in order to avoid non-trivial zero divisors, as exempli ed by the phenomenon (1 u) Pk2Z uk = 0). In our equation, F belongs to C[u; 1=u][q][[z]]. As the exponents of u range from 1 to +1, it is not legitimate to substitute u by our Puiseux series u(z), because this would lead to arbitrary negative and positive powers of z. We need to adapt the kernel method and to transform it into what we call a \bilateral kernel method": 1. Extract the positive part fu>0g. This gives the following equation fu>0g (1
1 zP (u)) X
! Wkuk
= z(1 In this expression, it is legitimate to substitute u by the c roots ui(z) of 1 z 0. Thus we get c new equations:
1 q) X fu>0g(P (u)uk)Wk 3. Extract [u0]. It gives one additional equation: [u0] (1 zP (u))
c X Wkuk k= d !! = 1: All these equations involve (a subset of) of the yet unknown c + d + 1 auxiliary functions W d; : : : ; Wc. This system fully allows to reconstruct the initial equation: W (the linear combination of the Wk's) is also a solution of the initial functional equation, which was a contraction in the space of formal power series C[q; u; 1=u][[z]], therefore this functional equation had a unique solution in this ring, which we thus identi ed as being W , the only candidate for it. This explains why the system of equations is of full rank: it has unique power series solutions Wk, all of them expressible as a quotient of polynomials having the roots as variable. (We say more on this shape in the full version of this article: it requires an excursion (sic) via the Schur function world!) So, in all cases, one gets a closed form for the generating function W (and the Wk's) in terms of the roots of the kernel. 0 = zq fu>0g P (u)uk Wk + zfu>0g P (u) W0 zP (u))uk Wk c Xfu>0g (1 k=1 2. Extract the negative part fu<0g. This leads to an equation in which it is legitimate to substitute u by the d roots vi(z) of 1 zP (u) such that jvi(z)j 1 for z 0. Thus we get d new equations: 0 =
c 1 zq Xfu<0g P (u)uk Wk + zfu<0g P (u) W0
1 X fu<0g (1 k= d zP (u))uk Wk ! ! ju=ui(z) ju=vi(z) : :
Now, the local analysis of these roots at their branching points, as done in [BF02] allows to get the Puiseux behaviour of the functions W (z; u; 1) and W0(z; u) near their dominant singularity (for any xed value of u, this dominant singularity is still the same as in [BF02]). It allows to rewrite locally these functions (for z ) in the framework of the schemes developed in [Wal16], on which we comment more in the next section.
Let us now illustrate this approach on basketball walks (walks with jumps 2; 1; 0; +1; +2, see [BKKKKNW17a]). This leads to the simple linear system: 8>W0 = 1 + z(W 1 + W 2 + W0 + W1 + W2) > >>>qzW 1 + (zu1 + z) W0 + zu12 + zu1 + z > <
qzW 1 + (zu2 + z) W0 + zu22 + zu2 + z >>>qzW1v13 + (v1 + 1) zW0v12 + (z > >:>qzW1v23 + (v2 + 1) zW0v22 + (z 1 W1 + zu13 + zu12 + zu1 + z 1 W1 + zu23 + zu22 + zu2 + z
Note that the Cardan/Ferrari{Bombelli formulas are perhaps nice for the eyes (arguably!), but they are not the right way to handle these generating functions from a computer algebra point of view. It is indeed better to work directly with symmetric functions of the roots ui's (and this advantageously also allows to handle the cases of degree > 4): this is an e cient algorithmic way to use the Newton relations/Vieta's formulas between the roots of the kernel. Using the expression with the small roots is also the way to follow the Puiseux behaviour of these symmetric functions, as we detail in the forthcoming full version.
What is more, the above linear system shows that it is now routine to get in a few minutes thousands of coe cients of our generating functions (e.g. via Newton iteration, in any computer algebra software). Here are the rst terms of the generating functions of x-axis crossings for walks: W (z; 1; q) =1 + 5 z + (2 q + 23) z2 + 2 q2 + 14 q + 109 z3+ 2 q3 + 16 q2 + 88 q + 519 z4 + 2 q4 + 18 q3 + 112 q2 + 504 q + 2489 z5+ 2 q5 + 20 q4 + 138 q3 + 700 q2 + 2776 q + 11989 z6+ 2 q6 + 22 q5 + 166 q4 + 930 q3 + 4150 q2 + 14896 q + 57959 z7+ 2 q7 + 24 q6 + 196 q5 + 1196 q4 + 5878 q3 + 23720 q2 + 78614 q + 280995 z8+ 2 q8 + 26 q7 + 228 q6 + 1500 q5 + 8004 q4 + 35518 q3 + 132264 q2 + 410046 q + 1365537 z9 + O z10 : What is the asymptotic behaviour of these polynomials in q? This is what we present in the next section.
Theorem 3.1 (Limit law for the local time at 0). For a walk with step set encoded by P (u), the limit laws for the local time at 0 depend on the drift P 0(1) of the walk (see Table 2):
Excursion Meander with drift < 0 Meander with drift 0 Walk with non-zero drift Walk with zero drift Bridge
Negative binomial distribution Negative binomial distribution Geometric distribution Geometric distribution Half-normal distribution H( )
Rayleigh distribution R( )
The parameters of these distributions are given in the proof and in Table 2: E.g. for bridges, the parameter of R( ) is = pP 00(1)=P (1), for walks, the parameter of H( ) is = =2pP (1)=P 00(1), where is the unique real positive value such that P 0( ) = 0.
What is more, if de ned, the limit laws for the the number of x-axis crossings are the same as the ones of the local time (with an appropriate new value for the distribution parameters).
Proof (Sketch). Thanks to the Puiseux expansions following from Theorem 2.1 and [BF02], it is possible to derive the limit laws: we then get a shape on which we can apply the results of Drmota and Soria [DS97] on the Rayleigh and Gaussian distributions, and of [Wal16] on the half-normal distribution. These distributions are depicted in Table 2. Details are omitted in this extended abstract.
In the cases of excursions and meanders the local time is equal to the number of returns to zero. The results for excursions were derived by Banderier and Flajolet in [BF02, Theorem 5]. What is more, all ingredients for the case of meanders are also given in this paper, and the result follows the same lines. However, due to the drift dependent number of meanders (compare [BF02, Theorem 4]) three regimes need to be considered, leading to two di erent limit laws: negative binomial and geometric.
Geom(p)
NB(m; p) H( )
R( ) Support
Mean Variance k 2 f0; 1; : : :g (1
p)kp 1 p
p 1 p p2
k 2 f0; 1; : : :g m+k 1 (1 k
p)kpm m(1 p)
p m(1 p) p2 q 2 2 exp
For the cases of walks and bridges the local time is equal to the number of returns to zero and the number of x-axis crossings. The results for the number of returns to zero of walks were derived by Wallner in [Wal16, Theorem 4.2] leading to a geometric or a half-normal distribution, depending whether the drift P 0(1) is non-zero or zero, respectively. As above, the result for the case of returns to zeros in bridges, follows also the same lines as the previous one and we omit the details in this extended abstract. In this case one uses the limit law of Drmota and Soria [DS97, Theorem 1] to prove the existence of a Rayleigh distribution.
It remains to consider the laws of x-axis crossings in the cases of bridges and walks. The proof of Theorem 2.1 gives access to closed forms of W0(z; q) and W (z; 1; q), which are the generating functions of bridges and walks where crossings of the x-axis are marked by q. In order to treat both at the same time we abbreviate them until the end of this proof by F (z; q). We want to apply either [DS97, Theorem 1] or [Wal16, Theorem 2.1]. Note that the technical conditions of these theorems are satis ed due to the closed forms in terms of small and large branches, and the fact, that due to the Weierstrass Preparation Theorem, the branches u1(z) and v1(z) (which are the real positive branches for z > 0 in the vicinity of 0, see Figure 2) satisfy the necessary conditions (compare the derivations in [Wal16]). In particular, as proven in [BF02], they satisfy a square root behaviour with the following expansion at z = : u1(z) = v1(z) = + s s 2 P ( ) p1
P 00( ) 2 P ( ) p1
P 00( ) z= + : : : ; z= + : : : : Due to the methods derived in [BW17], we may assume without loss of generality, that our model is aperiodic. For such aperiodic models (like e.g. Motzkin and basketball walks) there exists a unique singularity > 0 of F (z; 1).
Now, the key fact is that our generating functions F have locally the following behaviour 1 F (z; q) = g(z; q) + h(z; q)p1 z= ; for jq 1j < " and jz j < " with arg(z ) 6= 0 where " > 0 is some xed real number, and g(z; q) and h(z; q) are analytic functions. Additionally, one has here that g( ; 1) = 0. Finally, in the case of bridges we get gq( ; 1) < 0 and h( ; 1) 6= 0 yielding a Rayleigh law. Whereas, in the case of walks with zero drift we get gq( ; 1) = gqq( ; 1) = 0 and h( ; 1) = 0, but gz( ; 1) 6= 0 and hq( ; 1) 6= 1 giving a half-normal law. The respective parameters depend on the chosen step set.
with a square root behaviour. This In this article we showed how to derive the generating function for the local time at 0 of directed lattice paths. It completes the work of Banderier and Flajolet [BF02], who just handled the case of returns to zero of excursions. It is also extending the work of Feller [Fel68] and later Wallner [Wal16], who did the Dyck and Motzkin cases (for meanders/walks).
In order to solve the generating functions in the more general case, we used a mixture of the Omega operator and the kernel method. This leads to expressions from which we showed how to derive the limit law of the local time, for several models of constrained lattice paths. In the full version of this article, we give more closed form formulas, and we show that other parameters (like the number of \changes of signs", or the number of jumps from positive to negative altitude) can also be analysed using our approach, and that they satisfy similar limit laws.
These parameters are very natural for discrete random walks, it is interesting that it is not possible to analyse them via a Brownian motion approach: indeed a Brownian motion can be seen as the limit after a rescaling of the amplitude by pn and the length by n (see [Mar03]). This rescaling implies that (discrete) jumps crossing the abscissa would be of amplitude 0, and are therefore completely erased. It is therefore nice that analytic combinatorics can get the asymptotics of this \discrete local time", and the corresponding universal limit laws, while there are \invisible" via a Brownian motion approach. In a forthcoming article, we tackle further analysis of the height of discrete lattice paths; this allows to get the connection with the Brownian local time.
This work was started via collaboration funded by the SFB project F50 \Algorithmic and Enumerative Combinatorics" and the Franco-Austrian PHC \Amadeus", and ended during the postdoctoral position of Michael Wallner at the University of Paris Nord, in September-December 2017, thanks to a MathStic funding. Michael Wallner is currently supported by the Erwin Schrodinger Fellowship of the Austrian Science Fund (FWF): J 4162-N35. [APR01] [BD15] C. Banderier and M. Wallner. The kernel method for lattice paths below a rational slope. In: Lattice path combinatorics and applications, Developments in Mathematics Series. Springer, 2017. To appear. [MRSV99]
M. Bousquet-Melou and M. Mishna. Walks with small steps in the quarter plane. In: Algorithmic probability and combinatorics, volume 520 of Contemp. Math., pp. 1{39. Amer. Math. Soc., Providence, RI, 2010.
E. Csaki and I. Vincze. On some problems connected with the Galton-test. Matematikai Kutato Intezetenek Kozlemenyei. Magyar Tudomanyos Akademia, 6:97{109, 1961.
D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri. Underdiagonal lattice paths with unrestricted steps. Discrete Applied Mathematics, 91(1-3):197{213, 1999.
scheme for generating