The reflection-absorption model
for directed lattice paths
a
b
Cyril Banderier , Michael Wallner
a
Laboratoire d’Informatique de Paris Nord, UMR CNRS 7030, Université Paris Nord, France
b
E104 - Institute of Discrete Mathematics and Geometry, TU Wien
Asymptotic number of excursions
Lattice paths
Let en be number of excursions of length n. Then, the generating function of
excursions is of the kind
X
1
n
.
E (z) :=
en z =
1
−
zP
(u
(z))
0
1
n≥0
• Step set: S = {(1, b1), . . . , (1, bm )} ⊂ Z2
• n-step lattice path: Sequence of vectors (v1, . . . , vn ) ∈ S n
Probabilistic weights
• For S = {−c, . . . , d } define Π = {p−c , . . . , pd }, pi ∈ [0, 1] and
d
P
• Jump polynomial P(u) =
pi u
P
i pi = 1
i
i=−c
• P(1) = 1
The asymptotic number of excursions is given

−n
1


γρ1 −n 1 + O n ,
1 ρ
1
√
1
+
O
,
en ∼ κ π n1/2
n

−n

ρ
1
κ
√
1+O n ,
2 π(1−ρP0(τ ))2 n3/2
by
supercritical case: λ > 1,
critical case: λ = 1,
subcritical case: λ < 1.
The reflection-absorption model
Returns to zero
• Lattice: Z2+
• Altitude k 6= 0
• Weighted step set S
Pd
i
• P(u) =
p
u
i
i=−c
• Altitude k = 0
• Weighted step set S0
Pd0
i
• P0(u) =
p
u
i=0 0,i
Figure : Chosen steps depend on altitude
Definition
• An arch is an excursion of size > 0
whose only contact with the x-axis is
at its end point.
• A return to zero is a vertex of a path of Figure : An excursion with 3 returns to
zero
altitude 0 whose abscissa is positive.
Corresponding generating function
X
1
n k
E (z, u) :=
en,k z u =
1 − uzP0(u1(z))
n,k≥0
Reflection model: No loss of mass at 0: P0(1) = 1
Absorption model: Loss of mass at 0:
P0(1) < 1
Different constraints on the boundary
• Characteristic polynomial:
• Reflection model:
• Absorption model:
Excursion of length n having k returns to zero
en,k
P(Xn = k) := P(size = n, #returns to zero = k) =
en
Limit law for returns to zero of excursions
P(u) = pu + qu −1
P0(u) = u
P0(u) = p0u with p0 < 1
absolute value excursions,
1. In the supercritical case, i.e. λ > 1,
Xn − µn
√ , with
σ n
Dyck
bridges,
excursions,
path
uniform model
of bridges
1
6
1
6
1
6
1
1+p0/p+q0/q
p
1+p
p
p+p0
p0/p+q0/q
1+p0/p+q0/q
1
1+p
p0
p+p0
0
0
0
1
6
1
6
1
6
0
0
0
0
0
0
0
0
0
reflection m. absorption m.
µ = γ,
σ=γ
3
3
(α2ρ1
2
− 2) + γ (ρ1 + 2) − γ,
converges in law to a Gaussian variable N(0, 1).
2. In the critical case, i.e. λ = 1, the normalized
random variable √κ2n Xn, converges in law to a
−x 2/2
Rayleigh distribution (density: xe
):
κ
−x 2/2
.
lim P √ Xn ≤ x = 1 − e
n→∞
2n
3. In the subcritical case, i.e. λ < 1, the limit
distribution of Xn − 1 is the negative binomial
distribution NegBin(2, 1 − λ):
P(Xn − 1 = k) ∼ (k + 1)λk (1 − λ)2 .
Relevant constants
Let τ be the structural constant given by P 0(τ ) = 0, τ > 0, and let
ρ = 1/P(τ ) be the structural radius.
Conclusions
Let u1(z) be the unique solution of the kernel equation
• Similar results hold for the asymptotics of bridges and meanders,
1 − zP(u) = 0,
with limz→0 u1(z) = 0. Then, the equation 1 − zP0(u1(z)) = 0 has at most
one solution in z ∈ (0, ρ], which we denote by ρ1.
• Limit laws for other parameters like final altitude of meanders, or returns to
zero of bridges exist,
• Extensions to more general lattice path models are possible.
Additionally, we define the constants
α = (P0(u1(z)))0|z=ρ1 ,
1
γ= 2
,
αρ1 + 1
P0(τ )
λ=
.
P(τ )
α2 = (P0(u1(z)))00|z=ρ1 ,
s
P(τ ) 0
κ = ρ 2 00 P0(τ ),
P (τ )
References
[1] Banderier, C.; Flajolet, P.: “Basic analytic combinatorics of directed lattice paths”. Theoretical Computer
Science, 281, p. 37-80, 2002.
[2] Banderier, C.; Wallner, M.: “Some reflections on lattice paths”. Proceedings of the 25th Int. Conf. on Prob.,
Comb. and Asymptotic Methods for the Analysis of Algorithms (AofA’14), p. 25-36, 2014.
[3] Flajolet, P.; Sedgewick, R.: “Analytic Combinatorics”. Cambridge University Press, 2009.