4 Vertex of Maximal Exit Degree, Arrival Cells and BTS-Instants
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High Moments of Strongly Diluted Random Matrices
365
3.4.1 Reduction Procedures and Reduced Sub-walks
Given W2s , let t0 be the minimal instant of time such that
(i) the step st0 is the marked step of W2s ;
(ii) the consecutive to st0 step st0 C1 is non-marked;
(iii) W2s .t0 1/ D W2s .t0 C 1/.
If such t0 exists, we apply to the ensemble of steps S D fst ; 1 Ä t Ä 2s; st 2 W2s g a
reduction RP that removes from S two consecutive elements st0 and st0 C1 ; we denote
P
R.S/
D S0 . The ordering time labels of elements of S0 are inherited from those
of S.
The new sequence S0 can be regarded again as an even closed walk. We can apply
P Repeating this operation maximally
to this new walk the reduction procedure R.
possible number of times m, we get the walk
P m .W2s /;
WP2Ps D .R/
sP D s
m;
P m .S/ and say that
P D .R/
that we refer to as the strongly reduced walk. We denote S
P
R is the strong reduction procedure.
We introduce a weak reduction procedure RM of S that removes from S2s the pair
.st0 ; st0 C1 / in the case when the conditions (i)–(iii) are verified and
M
(iv) W2s .t0 / ¤ ˇ.
We denote by
M l .W2s /;
WM2Ms D .R/
sM D s
l
(39)
the result of the action of maximally possible number of consecutive weak
M D .R/
M l .S/. In what follows, we sometimes omit the
reductions RM and denote S
subscripts 2Ps and 2Ms. Regarding the example walk W16 (23), we observe that ˇM D ˛3
and that the strongly and weakly reduced walks coincide and are as follows,
WP8 D WM8 D .˛1 ; ˛2 ; ˛3 ; ˛5 ; ˛1 ; ˛2 ; ˛3 ; ˛5 ; ˛1 /:
M nS
P D S,
R we see that it represents a collection of subTaking the difference S
R D [j WR . j/ . Each sub-walk WR . j/ can be reduced by a sequence of the
walks, W
strong reduction procedures RP to an empty walk. We say that WR . j/ is of the Dycktype structure. It is easy to see that any WR . j/ starts by a marked step and ends by a
P between these two steps of WR . j/ . We
non-marked steps and there is no steps of W
. j/
R
say that W is the non-split sub-walk.
366
O. Khorunzhiy
L D Sn S
M is given by a collection
It is not hard to see that the collection of steps S
L D [k S
L .k/ represents a non-split Dyck-type sub-walk
L .k/ , each of S
of subsets S
WL .k/ ,
L D [k WL .k/ :
W
(40)
In this definition, we assume that each sub-walk WL .k/ is maximal by its length.
3.4.2 Arrival Instants and Dyck-Type Sub-walks Attached to ˇM
Given W2s , let us consider the instants of time 0 Ä t1 < t2 < : : : tR Ä 2s such that
for all i D 1; : : : ; R the walk arrives at ˇM by the steps of WM2Ms ,
W2s .ti / D ˇM
and sti 2 WM2Ms ;
i D 1; 2; : : : ; R:
(41)
We say that ti are the Mt-arrival instants of time of W2s . Let us consider a sub-walk
that corresponds to the subset SŒti C1;tiC1 D fst ; ti C 1 Ä t Ä tiC1 g Â S; we denote
this sub-walk by WŒti ;tiC1 . In general, we denote a sub-walk that is not necessary
even and/or closed by WŒt0 ;t00 also.
Let us consider the interval of time Œti C 1; tiC1 1 between two consecutive
M It can happen that W2s arrives at ˇM at some instants of time t0 2
Mt-arrivals at ˇ.
M We denote by Lt.i/ the maximal value of such t0 .
Œti C 1; tiC1 1, W2s .t0 / D ˇ.
Lemma 4 The sub-walk WŒti ;Lt.i/ coincides with one of the maximal Dyck-type sub0
walks WL .k / of (40).
Lemma 4 is proved in [10].
Let us consider a collection of all marked exit edges from ˇM performed by the
marked steps on the interval of time Œti ; Qt.i/ and denote this collection by L i . We say
M Or simply that
that L j represents the exit sub-clusters of Dyck type attached to ˇ.
L j are the exit sub-clusters of W2s . We denote their cardinalities by dj D j L j j. The
exit sub-clusters are ordered in natural way. To keep a unified description,
P we accept
the existence of empty exit sub-clusters; then we get equality D D RjD1 dj ; dj 0.
Clearly, any exit sub-cluster is attributed to a uniquely determined Mt-arrival instant
M
at ˇ.
Regarding the reduced walk WM2Ms of W2s (39), we can determine corresponding
M It is easy to show that TMsM is
Dyck path ÂM2Ms D Â.WM2Ms / and the tree TMsM D T .Â/.
a sub-tree of the original tree Ts D T .Â.W2s //. One can introduce the difference
TL D Ts ŸTMsM and say that it is represented by a collection of sub-trees TL . j/ .
Returning to the Catalan tree T .Â2s /, let us consider the chronological run over
it that we denote by RT . Then the Mt-arrival instant tl (41) determines the step $l of
RT . Also the corresponding vertex L l of the tree Ts is determined. It is clear that L l
are not necessarily different for different l.
High Moments of Strongly Diluted Random Matrices
367
The sub-trees TL .l/ are attached to L l and the chronological run over TL .l/ starts
immediately after the step $l is performed. We will say that these steps $l , 1 Ä
l Ä R represent the nest cells from where the sub-trees TL .l/ , 1 Ä l Ä L grow. It
is clear that the sub-tree Tl has dl
0 edges attached to its root %l and this root
coincides with the vertex L l . Returning to W2s , we will say that the arrival instants
M In the next sub-section, we describe a
of time Mtl represent the arrival cells at ˇ.
M
classification of the arrival cells at ˇ that represents a natural improvement of the
approach proposed in [12].
3.4.3 Classification of Arrival Cells at ˇM
Let us consider a walk W2s together with its reduces counterparts WP2Ps D WP and
WM2Ms D WM . Let ti denote a Mt-arrival cell (30). If the step sti of W2s is marked, then
M If the step st is non-marked and st 2
we say that ti represents a proper cell at ˇ.
i
i
M If the step st 2 WP is
R
M
P
W D W n W , then we say that ti represents a mirror cell at ˇ.
i
M
non-marked, then we say that ti represents an imported cell at ˇ.
R We denote by xi the
Let us consider I proper cells Rti such that sRti belongs to S.
corresponding marked instants, xi D Rti , 1 Ä i Ä I and write that xN I D .x1 ; : : : ; xI /.
It is easy to see that each proper cell xi can be attributed by a number 1 or 0 in
dependence of whether it produces a corresponding mirror cell at ˇM or not. We
denote this number by mi 2 f0; 1g and write that
MD
I
X
mi
iD1
and m
N I D .m1 ; : : : ; mI /. Clearly, M Ä I.
Regarding the strongly reduced walk WP2Ps , we denote by Ptk the proper cells such
P Corresponding to Ptk marked instants will be denoted by zk ,
that the steps sPtk 2 S.
1 Ä k Ä K. Then zNK D .z1 ; : : : ; zK / and the total self-intersection degree of ˇM is
M D I C K.
~.ˇ/
P there exists at least one pair of elements of S
P
Given W2s with non-empty set S,
0 00
0
00
P
denoted by .s ; s / such that s is a marked step of W2Ps , s is the non-marked one
P We refer to each pair of this kind as to the
and s00 follows immediately after s0 in S.
pair of broken tree structure steps of W2s or in abbreviated form, the BTS-pair of
W2s . If 0 is the marked instant that corresponds to s0 , we will simply say that 0 is
the BTS-instant of W2s [12].
Regarding the strongly reduced walk WP , let us consider a non-marked arrival
step at ˇM that we denote by sN D sNt . Then one can find a uniquely determined marked
P with 0 C 1 Ä t Ä Nt are the non-marked ones.
instant 0 such that all steps st 2 S
00
Let us denote by t the instant of time of the first non-marked step sNt00 2 SO of this
series of non-marked steps. Then .st0 ; st00 / with t0 D 0 is the BTS-pair of W2s that
corresponds to Nt. We will say that Nt is attributed to the corresponding BTS-instant 0 .
368
O. Khorunzhiy
It can happen that several arrival instants Nti are attributed to the same BTS-instant
0
. We will also say that the BTS-instant 0 generates the imported cells that are
attributed to it.
M As it is said above,
Let us consider a BTS-instant such that W2s . / D ˇ.
there are K such marked instants denoted by zk , 1 Ä k Ä K. We refer to such
BTS-instants as to the local ones. Assuming that a marked BTS-instant zk generates
.k/
.k/
fk0 0 imported cells, we denote by '1 ; : : : ; 'f 0 the positive numbers such that
k
W2s .
zk
C
l
X
.k/
'j / D ˇM
for all 1 Ä l Ä fk0 :
(42)
jD1
If for some kQ we have fkQ0 D 0, then we will say that zkQ does not generate any imported
M We denote 'N .k/ D .' .k/ ; : : : ; ' .k/
cell at ˇ.
0 /.
1
fk
Let us consider a BTS-instant that generates imported cells at ˇM and such that
M We denote these BTS-instants by yj , 1 Ä j Ä J a say that yj is a remote
W2s . / ¤ ˇ.
M Assuming that a marked BTS-instant yj generates
BTS-instant with respect to ˇ.
. j/
. j/
00
00
0, we denote by j ; 1 ; : : : ; f 00 the positive numbers
fj C 1 imported cells, fj
j
such that W2s .
yj
C
W2s
j/
yj
D ˇM and
C
j
C
k
X
!
. j/
l
D ˇM
for all 1 Ä k Ä fj00 :
(43)
lD1
In this case we will say that the first arrival at ˇM given by the instant of time yj C
M
M
j represents the principal imported cell at ˇ. All subsequent arrivals at ˇ given
M We will use denotations yN J D
by (41) represent the secondary imported cells at ˇ.
. j/
. j/
N
.y1 ; : : : ; yJ / and J D . 1 ; : : : ; J /. We also denote N . j/ D . 1 ; : : : ; f 00 /.
j
We see that for a given walk W2s , the proper, mirror and imported cells at its
vertex of maximal exit degree are characterized by the set of parameters, .Nx; m/
N I,
.Nz; ˚; fN 0 /K , where ˚K D .'N .1/ ; : : : 'N .K/ /, fNK0 D . f10 ; : : : ; fK0 / and .Ny; N ; «; fN 00 /J , where
«J D . N .1/ ; : : : ; N . j/ /, fNJ00 D . f100 ; : : : ; fJ00 /. We also denote
F0 D
K
X
kD1
fk0
and F 00 D
J
X
fj00 :
jD1
Summing up, we observe that the vertex ˇM with the self-intersection degree
M D I C K has the total number of cells given by R D I C M C K C J C F,
~.ˇ/
where I is the number of proper cells from the Dyck-type parts, M is the number
of corresponding mirror cells, K is the number of local BTS-instants and J is the
number of remote BTS-instants, F represents the number of imported cells at ˇM
High Moments of Strongly Diluted Random Matrices
369
generated by the local and remote BTS-instants, F D F 0 C F 00 . In what follows, we
denote the family of the parameters described above by
«
˚
PR D .Nx; m/
N I ; .Ny; N ; «; fN 00 /J ; .Nz; ˚; fN 0 /K :
(44)
3.5 Proof of Theorem 2
Q 2s .D/ that
We are going to estimate the number of walks in the family of walks W
have a vertex of maximal exit degree D. We rewrite (28) in the following form
ZQ2s .n; / D
s
X
X
˘a .W2s / ˘b .W2s / jCW2s j ;
DD1 W2s 2W
Q 2s .D/
Q 2s .D/, we
where jCW2s j is given by (24). To estimate the number of elements in W
have to consider a kind of color diagrams that have a separate vertex vM attributed
by the parameters from the family PR , namely by xN I and zNK . Also we have to
incorporate into the diagram description the parameters yN J . Thus we get a new type
of color diagrams that we are going to determine.
3.5.1 Color Diagrams with a Vertex of Maximal Exit Degree
Let us consider a vertex vM and attach to it I C K edge-boxes. We denote by hvM I;K is a
realization of the values of marked instants that fill these boxes. Given N ; pN and qN , we
consider a realization of the corresponding color diagram hG .c/ . N ; pN ; qN /is and point
out J edge-boxes that will provide the marked instants yN . Joining such a realization
.c/
.b/
with chosen J edge-boxes hGyN . N ; pN ; qN /is with hvM I;K is , we get a realizations of the
diagram we need,
.c/
.c/
hGMxN;Nz;Ny . N ; pN ; qN /i.b/
M I;K is ] hGJ . N ; pN ; qN /i.b/
M ] G .c/ i.b/
s D hv
s D hv
s :
The last equality of the formula presented above introduces a denotation for a
realization of the diagram we consider.
The number of different realizations of the color diagram G .c/ . N ; pN ; qN / is estimated by the right-hand side of (34). Regarding realizations hvM I;K is , we can write
that
jhvM I;K is j Ä
sICK
2ICK ;
.I C K/Š
(45)
where the last factor gives the upper bound for the choice of K elements among ICK
ones to be marked as the values of zNK . The vertex ˇM of the walk can be attributed by
370
O. Khorunzhiy
the weight
8
V2 ;
M D 1;
ˆ
if ~.ˇ/
ˆ
ˆ
2 2.ICK/ 4
V U
; if ˇM is an r-vertex;
M ˘b .ˇ/
M D
˘a .ˇ/
n2 ICK 2 2
ˆ
ˆ
ˆ
1
2.ICK/ 2
: ICK
; if ˇM is a p-vertex or a q-vertex.
1 V2 U
n
(46)
In the first and in the third cases of (46), at least one blue r-vertex is necessarily
present in G .c/ . N ; pN ; qN /.
Regarding hG .c/ . N ; pN ; qN /is , one can
Pchoose J edge-boxes to be labeled as the
values of the realization hNyi among skD2 .k 1/ k D k N k1 edges only. This is
because the first arrival to a vertex cannot be the marked BTS-instant. The number
of ways to choose J ordered places among k N k1 ordered edges can be estimated as
follows,
!
k N k1
1
k N kJ1
Ä J exp fh0 k N k1 g ;
(47)
Ä
J
JŠ
h0
where h0 > 1 is a constant.
3.5.2 Exit Sub-clusters and Cells at ˇM
The maximal exit degree of a walk W2s 2 W2s .D/ can be represented as follows,
M / that
P CD
R C D,
L where D
P is the number of marked edges of the form .ˇ;
DDD
R represents the exit edges that belong
belong to the strongly reduced walk WP (39), D
P D FCJ and that F Ä K [12] (see also Lemma 12
to WR D WM n WP . It is known that D
R D M. Taking into account that M Ä I, we can write
of [9]). Also we observe that D
that
L DD
D
M
F
J
D
I
K
J:
(48)
N 2s / belong to the exit sub-clusters of the DyckL edges of E.W
The remaining D
.k/
M They are distributed among R arrival cells at
L
type sub-walks W (40) attached to ˇ.
P
M
N
L
L
L
ˇ. We denote by d D .d1 ; : : : ; dR / a particular distribution such that RlD1 dL l D D.
.c/ .b/
The number of cells R depends on hG is , Âs and . However, the inequalities
used to get (48) show that
R D I C K C M C F C J Ä 2I C 2K C J D R :
(49)
High Moments of Strongly Diluted Random Matrices
371
L C R Ä D C R and we deduce from (49)
Then the first relation of (48) implies that D
that
!
!
X
L CR 1
1
D
DCR
:
1D
Ä
1
R
R 1
L
dN R ;jdN R jDD
Elementary analysis shows that if D
!
DCR
1
1
Ä hR0 sup R
R
1
R 2 h0
1
2, then
!
DCR
1
eD
; h0 > e:
exp
Ä h2IC2KCJ
0
R
1
h0
(50)
Indeed, using the standard estimates
p
p n Án
n Án
2 n
Ä nŠ Ä e n
;
e
e
n
1;
we can write that
1 e
1 .D C m/Š
Ä m
hm
DŠ
mŠ
h
0
0 2
r
DCm
mD
m ÁD
1C
D
Ã
Ã
Â
Â
D m
D m
em
1C
Ä m 1C
;
m
h0
m
where we take into account that DCm Ä 2mD. Then the last relation of (50) follows.
Now we are ready to perform the estimates that prove Theorem 2.
3.5.3 Exponential Estimates and ZQ2s
In this subsection we estimate the contribution of the non-tree type walks ZQ2s and
prove relation (29) with the help of computations that are very similar to those used
in the pioneering papers by Ya. Sinai and A. Soshnikov. The following statement
can be regarded as the principal result of the method.
.b/
Lemma 5 Given D, a realization of the color diagram hvM ] G .c/ is and a rule ,
let us consider a family of walks W2s .D; hvM ] G .c/ is ; / such that the vertex of the
maximal exit degree given by vM has D exit edges of the form .v;
M i /, i D 1; : : : ; D.
Then
jW2s .D; hvM ] G .c/ is ; /j Ä 2jNqj DjNpj eÁ h20
where Á D ln.4=3/.
ICKCJ
e
ÁDCeD=h0
ts ;
(51)
372
O. Khorunzhiy
We prove Lemma 5 in Sect. 5. The walks we consider are of the non-tree type
and therefore contain at least one blue r-vertex v0 . Let us divide the sum ZQ2s in two
parts in dependence whether v0 D vM or v0 Ô v,
M
.1/
.2/
ZQ2s .n; / D ZQ2s C ZQ2s ;
(52)
respectively. Then we can write that
.1/
ZQ2s
D
!?
s
s X
X
s
Y
X
DD1 JD0
kD2 rk ;pk ;qk
X
X
.c/
v]G
M
s
.c/ i
hv]G
M
s
X
jCW2s j
I;KW ICK 1
X
W2s 2W2s
˘a;b .W2s /;
.c/ i
.D;hv]G
M
(53)
s/
P
where
Pthe star means that the values of rk ; pk and qk are such that .k 1/ k J
and rk 1. The first sum of the second line of (53) takes into account the choice
of the J places in G .c/ to be marked as the edge-boxes of values yj [see also (47)];
the second sum is performed over all possible realizations of the diagram vM ] G .c/
obtained with the help of the values from f1; : : : ; sg (see (34) for example).
Using relations (27), (37), (45) and (46), we deduce from (53) that
.1/
ZQ2s Ä
s
s
X
X
X
DD1 JD0 ICK
s
Y
X
1
.2s/ICK
.2.I C K//ICK h2.ICK/CJ
eÁ.ICJCK/
0
.I C K/Š
!?
eh0 kN k1
kD2 rk ;pk ;qk
e
ÁDCeD=h0
ts nsC1
V2 U 2.ICK 1/
n ICK 1
Â
V22
n2
1
rk Š
Â
.2k/k sk
.k 1/Š
.kN k1 C.ICK
Ãrk Â
V2 U 2
n
Ãrk
1
pk Š
Â
.2k/k Dsk 1
.k 2/Š
Ãpk
1
qk Š
Â
.2k/k 2sk 1
.k 2/Š
Ãqk
1//
Ãpk Cqk Â
U2
Ã.k
2/
k
Â
V2
n
Ãs
kN k .ICK/
;
P
where we denoted k N k D skD2 k k .
Let us consider a constant [cf. (32)]
(
2k
.2k/k=.k 1/
;
sup
C2 D max sup
1/Š/1=k k 2 ..k 2/Š/1=.k
k 2 ..k
and denote
B D C2 h0 eh0 CÁ D 4C2 h0 eh0 =3;
)
1/
(54)
High Moments of Strongly Diluted Random Matrices
373
where h0 > e will be determined below. Remembering that s D
from (54) the following inequality,
.1/
ZQ2s
Ä
V2s
s
X
e
ÁDCeD=h0
n ts
O2
2sB 2BU
ÁICK
BUO 2 s2 O 2 k
.BU /
n
!rk
2
1
s
Y
X
kD2 rk ;pk ;qk
JD0 ICK 1
DD1
If
s
X
X
, we can deduce
!?
1
rk Š
Ápk 1
Áqk
1
D.BUO 2 /k 1
2.BUO 2 /k 1 :
pk Š
qk Š
(55)
is such that
2BUO 2 Ä 1;
(56)
then (55) implies inequality
Â
2Bs2
.1/
ZQ2s Ä 4Bs3 exp
n
Ã
1
O2
e4BU nts V2s
1
X
n
exp
o
Á C 2BUO 2 C e=h0 D :
DD1
(57)
Remembering that Á D ln.4=3/ > 0:28, we see that if
3C2 U 2 h0 eh0
V2
C
e
Ä 0:28
h0
(58)
then
.1/
ZQ2s D O.nts V2s s5 =n/ D o.nts V2s /
in the limit .n; s; / ! 1 (4). Clearly, the choice of h0 and
h0 D 4e and
Ä
V2
4eC1
400e
C2 U 2
(59)
such that
(60)
makes (56) and (58) valid. Let us note that more detailed analysis of the walks with
maximal exit degree D show that the factor s3 in the right-hand side of (57) could
be eliminated. However, in the present paper we do not aim the maximal rate of n
and therefore the upper bound (57) is sufficient for our purposes.
.2/
Let us consider the second term of (52). The sub-sum ZQ2s can be estimated from
above by the expression given by the right-hand side of (54), where the sum over I; K
is performed over the range I CK 2 and the weight factor V2 U 2.ICK 1/ =.n ICK 1 /
is replaced by V22 U 2.ICK 2/ =.n2 ICK 2 / [see relation (47)] and where the condition
P
1 is omitted.
k rk
374
O. Khorunzhiy
Then we can write that
.2/
ZQ2s Ä nts V2s
s
X
e
s
s
X
X 2s2 B
Y
X 1
.2BOu2 /ICK 2
n
rŠ
JD0 ICK 2
kD2 rk ;pk ;qk k
!rk
Ápk 1
Áqk
1
k 2
D.BUO 2 /k 1
2.BUO 2 /k 1 :
/
pk Š
qk Š
ÁDCeD=h0
DD1
BUO 2 s2 O 2
.BU
n
If (60) is true, then we get the following upper bound
4
4s B 2BUO 2
.2/
e
ZQ2s Ä nts V2s
n
1
X
n
exp
o
Á C 2BUO 2 C e=h0 D :
DD1
Then
.2/
ZQ2s D O.nts V2s s4 =n/ D o.nts V2s /
(61)
under conditions of Theorem 1. Combining this estimate with the estimate of
.1/
ZQ2s (59), we get (29). Theorem 2 is proved.
4 Tree-Type Walks and .2 ; 4? /-Walks
O 2s of tree-type walks and separate it into two nonLet us consider the family W
intersecting subsets,
P 2s t W
R 2s ;
O 2s D W
W
P 2s contains the walks W2s such that their weights have the factors V2 D 1
where W
and V4 only and the graph gN .W2s / is such that the V4 -edges do not share a vertex in
?
.2;4? /
P 2s and say that if W2s 2 W.2;4 / ,
common. We also denote this set by W2s
DW
2s
then this W2s is a tree-type .2; 4? /-walk. We denote
.n; /
ZP2s D
X
˘a;b .W2s / jCW2s j ;
.2;4? /
W2s 2W2s
.n; /
ZR2s D
X
˘a;b .W2s / jCW2s j
R 2s
W2s 2W
.n; /
.n; /
.n; /
and ZO 2s D ZP2s C ZR2s . Let us point out that two following relations are true,
jCW2s j D njV.W2s /j .1 C o.1//; n ! 1
and
jCW2s j Ä njV.W2s /j ;
where V.W2s / is the ensemble of vertices of the graph gN .W2s /.
(62)
High Moments of Strongly Diluted Random Matrices
375
Theorem 3 Under conditions of Theorem 1, the following upper bounds are true
1 P .n; /
Z2s Ä 4 expf16V4 g
/!1 nts
lim sup
.n;s;
(63)
and
lim sup
.n;s; /!1
for all 0 <
Ä
0
D
0 .U/
0 .U/
nts
.n; /
ZR2s Ä C expf16V4 g;
and C
D
1
411 U 2
(64)
C0 D C0 .U/, where
and C0 .U/ D 3 416 U 6 :
(65)
Remark Theorem 3 can be proved under conditions of Theorem 1 with (5) replaced
by much less restrictive condition on the probability distribution of aij to be such
that all its moments exist and are bounded as follows,
V2C2k Ä kŠ V2 bk0 ;
k D 2; 3; : : :
(66)
with given b0 > 0 (see also [13]). In this case the constants of (65) should be
replaced by
0
0 .b0 /
D
1
3 219 b0
and
C00 .b0 / D 3 416 b20 ;
(67)
respectively, where we assumed that (66) holds with V2 D 1.
To describe the general structure of the tree-type walk, let us introduce several
auxiliary notions. Regarding a sub-walk of 2a steps W2a and its graph gN .W2a /, let
us denote by % the ensemble of the multiple edges of gN that make a connected
component attached to the root %. If the graph gN .W2a / has no other multiple edges
than those of % and the first step and the last step of W2a are performed along the
.1/
edges of % , we say that W2a is the element of the block of the first level Ba . /,
D %,
W2a D Ba 2 B.1/
a . /:
We will say also that W2s by itself is a block of the first level, when no confusion
can arise.
.2/
We say that a walk W2b is a block of the second level, W2s D Bb , if it starts and
ends with the steps along the root component of multiple edges % and in W2b there
exists at least on sub-walk W2a0 that is the block of the first level.