We now choose k
def
= ⌊c ln n⌋ + 1 with c
def
=
1 3 ln1η
and set δ
def
= c ln b. Since there are b
k
n
δ
increasing paths of length k, we get, for n large enough,
P ¦
[ ˜
u
1
, . . . , ˜ u
n
] contains less than n
δ
distinct increasing sub-paths of same length ©
≤ P [
˜ u
1
, . . . , ˜ u
n
] does not contain all increasing sub-paths of length k} ≤ b
k
1 − η
k ⌊nk⌋
≤ e
− p
n
. Thus, if V
n
denotes the number of distinct vertices visited by the cookie random walk X up to time n
, we have proved the lower bound:
P {V
n
≤ n
δ
} ≤ e
− p
n
. Choosing ν max1δ, 2, we conclude that, for all n large enough,
P
{γ
n
≥ n
ν
} = P{V
⌊n
ν
⌋
≤ n} ≤ P{V
⌊n
ν
⌋
≤ ⌊n
ν
⌋
δ
} ≤ e
−
p
⌊n
ν
⌋
≤ e
−n
.
6.2 Example of a transient walk with sub-linear growth
In this section, we prove Proposition 1.9 whose statement is repeated below.
Proposition 6.5. Let X be a C = p
1
, p
2
, 0, 0 ; q cookie random walk with q ≥ bb +1 and p
1
, p
2
0 such that the largest positive eigenvalue of the matrix P
1
def
=
p
1
b
+
p
1
p
2
b
−
2p
1
p
2
b
2
p
1
p
2
b
2
p
1
+p
2
b
−
2p
1
p
2
b
2
p
1
p
2
b
2
is equal to 1b such a choice of p
1
, p
2
exists for any b ≥ 2. Then, X is transient since q ≥ bb + 1
yet lim inf
n →∞
|X
n
| n
= 0. Proof.
For this particular cookie environment, it is easily seen that the cookie environment matrix P has three irreducible classes {0}, [1, 2], [3, ∞ and takes the form
P =
1
P
1
∗
infinite class
.
where P
1
is the matric given in the proposition. By hypothesis, the spectral radius of the irreducible class [1, 2] is 1b, therefore, the branching Markov chain L starting from ℓo ∈ [1, 2] dies out
1658
almost surely the restriction of L to [1, 2] is simply a critical 2-type branching process where each particle gives birth to, at most, 2 children. In particular, the quantity
Λ
def
= X
x ∈T
ℓx
is P
i
almost surely finite for i ∈ {1, 2}. Moreover, one can exactly compute the generating functions
E
i
[s
Λ
] for i ∈ {1, 2} using the recursion relation given by the branching structure of L. After a few lines of elementary but tedious calculus and using a classical Tauberian theorem, we get the
following estimate on the tail distribution of Λ:
P
1
{Λ x} ∼ C
p x
, for some constant C 0 depending on p
1
, p
2
alternatively, one can invoke Theorem 1 of [9] for the total progeny of a general critical multi-type branching process combined with the characterization
of the domain of attraction to a stable law. As in the previous section, let γ
n n
≥0
denote the increasing sequence of times where the walk visits a new site as defined in Lemma 6.4 and define two interlaced sequences S
i i
≥0
and D
i i
≥0
in a similar way as in the proof of Proposition 6.2 only the initialization changes:
¨ S
= 0 D
= inf{n 0, X
n
= X
n −1
= o}, and by induction, for k ≥ 1,
¨ S
k
= inf{γ
n
, γ
n
D
k −1
}, D
k
= inf{n S
k
, X
n
=
←
X
S
k
}. Since, L starting from ℓ0 = 1 dies out almost surely, the walk crosses the edge from the root to
the root at least once almost surely. Therefore, D is almost surely finite. Using the independence of
the cookie random walk on distinct subtrees, it follows that the random variables sequences S
i i
≥0
, D
i i
≥0
are all finite almost surely. Moreover, recalling the construction of L, it also follows that the sequence of excursion lengths R
i i
≥0
def
= D
i
− S
i i
≥0
is a sequence of i.i.d. random variables,
distributed as the random variable 2Λ − 1 under P
1
. We also have the trivial facts: • For all k 0, |X
D
k
| = |X
S
k
| − 1. • The walk only visits new vertices of the tree during the time intervals [S
k
, D
k
]
k ≥0
. Thus, the number of vertices visited by the walk at time S
k
is smaller than 1 + P
k −1
i=
R
i
. In particular, we have |X
S
k
| ≤ 1 + P
k −1
i=
R
i
. • For all k ≥ 0, D
k
≥ P
k i=
R
i
. Combining these three points, we deduce that, for k ≥ 1,
|X
D
k
| D
k
≤ P
k −1
i=
R
i
P
k i=
R
i
= 1 − R
k
P
k i=
R
i
.
1659
Since R
i i
≥0
is a sequence of i.i.d random variables in the domain of normal attraction of a positive stable distribution of index 12, it is well known that
lim sup
k →∞
R
k
P
k i=
R
i
= 1 a
.s. this result follows, for instance, from Exercise 6, p.99 of [8] by considering a subordinator without
drift and with Lévy measure Λ. We conclude that
lim inf
n →∞
|X
n
| n
≤ lim inf
k →∞
|X
D
k
| D
k
= 0 a
.s.
7 Computation of the spectral radius
In this section, we prove Theorem 1.4 and Proposition 1.8 by computing the maximal spec- tral radius λ of the cookie environment matrix P. Recall that the irreducible classes of P are
{0}, [l
1
, r
1
], . . . , [r
K
, ∞ and that P
k
denotes the restriction of P to [l
k
, r
k
] [l
k
, ∞ for k = K. Denoting by λ
k
the spectral radius of P
k
, we have, by definition: λ = maxλ
1
, . . . , λ
K
. Since the non negative matrices P
1
, . . . , P
K −1
are finite, their spectral radii are equal to their largest eigenvalue. Finding the spectral radius of the infinite matrix P
K
is more complicated. We shall make use on the following result.
Proposition 7.1. Let Q = qi , j
i , j≥1
be an infinite irreducible non negative matrix. Suppose that there exists a non-negative left eigenvector Y = y
i i
≥1
of Q associated with some eigenvalue ν 0 i.e.
t
Y Q = ν
t
Y .
39 Assume further that, for all
ǫ 0, there exists N ≥ 1 such that the finite sub-matrix Q
N
= qi, j
1≤i, j≤N
is irreducible and the sub-vector Y
N
= y
i 1≤i≤N
is ν − ǫ super-invariant i.e
t
Y
N
Q
N
≥ ν − ǫ
t
Y
N
. 40
Then, the spectral radius of Q is equal to ν.
Remark 7.2. By symmetry, the proposition above remains unchanged if one considers a right eigen-
vector in place of a left eigenvector. Let us also note that Proposition 7.1 does not cover all possible cases. Indeed, contrarily to the finite case, there exist infinite non negative irreducible matrices for
which there is no eigenvector Y satisfying Proposition 7.1.
Proof. On the one hand, according to Criterion I of Corollary 4.1 of [23], the spectral radius λ
Q
of Q
is the smallest value for which there exists a non negative vector Y 6= 0 such that
t
Y Q ≤ λ
Q t
Y .
1660
Therefore, we deduce from 39 that ν ≥ λ
Q
. On the other hand, the matrix Q
N
is finite so that, according to the Perron-Frobenius Theorem, its spectral radius is equal to its largest eigenvalue λ
Q
N
and is given by the formula λ
Q
N
= sup
x
1
,...,x
N
min
j
P
N i=
1
x
i
qi , j
x
j
where the supremum is taken over all N -dimensional vectors with strictly positive coefficients c.f. 1.1 p.4 of [19]. In view of 40, we deduce that λ
Q
N
≥ ν − ǫ. Furthermore, when Q
N
is irreducible, Theorem 6.8 of [19] states that λ
Q
N
≤ λ
Q
. We conclude that λ
Q
≤ ν ≤ λ
Q
+ ǫ.
7.1 Preliminaries
