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4 Vertex of Maximal Exit Degree, Arrival Cells and BTS-Instants

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

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

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

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.

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